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A  TEXT-BOOK 

OF 

PHYSICS 


A  TEXT-BOOK  OF  PHYSICS 

A.  WILMER  DUFF,  Editor 


MECHANICS  AND  SOUND.  BY  A.  WILMER 
DUFF,  D.  Sc.,  Professor  of  Physics,  Worcester 
Polytechnic  Institute,  Worcester,  Massachusetts. 

WAVE  MOTION  AND  LIGHT.  BY  E.  PERCIVAL 
LEWIS,  PH.  D.,  Professor  of  Physics,  University 
of  California,  Berkeley,  California. 

HEAT.  BY  CHARLES  E.  MENDENHALL,  PH.  D., 
Professor  of  Physics,  University  of  Wisconsin, 
Madison,  Wisconsin. 

ELECTRICITY  AND  MAGNETISM.  BY  ALBERT 
P.  CARMAN,  D.  Sc.,  Professor  of  Physics,  Uni- 
versity of  Illinois,  Urbana,  Illinois. 

CONDUCTION  OF  ELECTRICITY  THROUGH 
GASES  AND  RADIO-ACTIVITY.  BY  R.  K. 
McCLUNQ,  D.  Sc.,  F.  R.  S.  C.,  Assistant  Pro- 
fessor of  Physics,  University  of  Manitoba,  Winni- 
peg, Manitoba. 


SCIENCE  SERIES 


A  TEXT-BOOK 

OF 

PHYSICS 


EDITED  BY 

A.   WILMER  DUFF 


CONTRIBUTORS 

A    WILMER  DUFF  ALBERT  P.  CARMAN 

E.  PERCIVAL  LEWIS  R.  K.  McCLUNG 

CHARLES  E.  MENDENHALL 


FOURTH  EDITION  REVISED 
WITH  609  ILLUSTRATIONS 


PHILADELPHIA 

P.   BLAKISTON'S   SON   &  CO 

1012  WALNUT   STREET 


I9Q&,  BT  P.  BLAOSTOK*S  SON  &  Co. 
BT  P.  I**A  •*!•«»*  Soar  &  Co. 

^IO   AND    IQII. 
BT  P.   BlAXISTCttf*S   SOU  &   CO. 

MAT,  1915- 

,    1916,  BT  P.    BLAMSK«"S  SOK  ft  Co 


PREFACE  TO  THE  FOURTH  EDITION 


-.:.-.   ----•-.    .:----  ::.-.  -..-..'..-:    -.-=-. 


EXTRACTS  FROM  PREFACE  TO  FIRST  EDITION 


The  preparation  of  a  work  of  this  grade  by  the  collaboration 
of  several  writers  is  a  somewhat  novel  undertaking,  and  some 
explanation  of  its  genesis  will  not  be  out  of  place.  It  represents 
the  attempt  of  seven  experienced  teachers  of  college  physics  to 
prepare  a  text-book  that  would  be  more  satisfactory  to  all  of 
them  than  any  existing  one.  It  was,  of  course,  hoped  that  such 
a  book  would  also  prove  acceptable  to  other  teachers.  It  seemed 
to  the  writers  that  there  was  a  need,  and  there  would  be  a  place, 
for  a  work  prepared  in  this  way. 

One  or  two  remarks  as  to  the  character  of  the  book  may  be 
permitted.  It  will  in  general  be  found  that  the  writers,  while 
aiming  first  of  all  at  clearness  and  accuracy,  have  preferred  terse- 
ness to  diffuseness.  Repetition  and  amplification  are  desirable 
in  a  lecture.  In  a  printed  statement,  which  may  be  reread  and 
weighed  until  mastered,  they  often  discourage  thought;  and  a 
teacher  of  Physics  might  well  begin  his  instruction  with  the  words 
of  Demosthenes,  "In  the  name  of  the  gods  I  beg  you  to  think." 
The  writers  have  endeavored  to  present  their  subjects  simply  and 
directly,  avoiding,  on  the  one  hand,  explanations  obvious  to  any 
student  of  fair  capacity,  and,  on  the  other  hand,  subtle  distinc- 
tions and  discussions  suited  to  more  advanced  courses.  Some 
may  find  the  material  included  in  the  book  too  extensive  for  a 
single  course.  If  so,  a  briefer  course  can  be  arranged  by  omitting 
all  of  the  parts  in  small  print  together  with  as  much  of  those  in 
large  print  as  may  seem  desirable.  There  may  seem  to  be  some 
duplication  of  topics  in  the  work  of  two  contributors.  In  such 
cases  (which  are  very  few),  it  will  be  found  that  the  treatment  is 
from  different  points  of  view,  appropriate  to  the  respective  sub- 
divisions of  the  subject. 

THE  EDITOR. 

WORCESTER,  MASS. 

vii 


CONTENTS 


MECHANICS  AND  PROPERTIES  OF  MATTER 

PAGE 

INTRODUCTION •  .    .  1 

MECHANICS. 

Displacements .    .    . 8 

^Velocity 11 

Acceleration 18 

Force  and  Mass 26 

Work  and  Energy 40 

Rotation 51 

Center  of  Mass " .    ....  56 

Moments .  62 

Resultant  of  Forces 68 

Forces  in  Equilibrium 73 

Periodic  Motions 78 

Friction 88 

Simple  Machines 93 

Gravitation 99 

Units  and  Dimensions 104 

PROPERTIES  OP  MATTER. 

Constitution  of  Matter 106 

Properties  of  Solids ;    .    ., 109 

Properties  of  Fluids 120 

Liquids . 132 

Molecular  Properties 139 

Gases . 148 

References  and  Problems 163 

WAVE  MOTION 

Types  of  Waves 174 

Composition  of  Simple  Harmonic  Motions 177 

Waves  due  to  Simple  Harmonic  Motions 181 

Superposition  and  Interference  of  Waves 184 

Velocity  of  Waves 186 

Reflection  of  Waves.     Stationary  Waves 188 

Refraction  of  Waves 193 

Ripples .  • 193 

References  and  Problems 197 

ix 


x  CONTENTS 

HEAT 

PA.QB 

Introduction 199 

Thermometry 202 

Expansion 213 

Calorimetry 226 

Change  of  State 240 

Convection  of  Heat 262 

Conduction  of  Heat 264 

Radiation 269 

Conservation  of  Energy 277 

Thermodynamics 281 

References  and  Problems 302 

ELECTRICITY  AND  MAGNETISM 

MAGNETISM. 

Magnets  and  Magnetic  Fields 309 

Measurements  of  Magnetic  Fields 323 

Earth's  Magnetic  Field 329 

ELECTROSTATICS. 

Electrification 334 

Theories  of  the  Nature  of  Electricity 338 

Electric  Fields  of  Force 343 

Potential 349 

Static  Electrical  Machines 356 

Capacity 359 

Atmospheric  Electricity 370 

ELECTROKINETICS. 

The  Electric  Current 371 

Magnetic  Field  of  the  Electric  Current 374 

Measurements  of  Currents.     Galvanometers 384 

Electromotive  Force  and  Resistance 390 

Heating  by  Electric  Current 402 

Electrolysis 405 

Primary  and  Secondary  Cells 411 

Thermoelectricity 419 

ELECTROMAGNETS  AND  MAGNETIC  INDUCTION. 

Electromagnets 424 

Magnetization  Curves 426 

Permeability  and  Hysteresis .  430 

ELECTROMAGNETIC  INDUCTION. 

Induced  Currents 437 

Self-induction 445 

Dynamo-electric  Machines 453 

Electrodynamics 463 

Electric  Oscillations  and  Waves   .  .  470 


CONTENTS  xi 

PAOB 

Dimensions  of  Electrical  Units 480 

References  and  Problems 482 

CONDUCTION  OF  ELECTRICITY  THROUGH  GASES  AND 
RADIOACTIVITY 

Conduction  of  Electricity  through  Gases 489 

Cathode  and  Rontgen  Rays 490 

lonization  of  Gases 496 

Radioactivity 502 

Rays  Emitted 504 

Emanation  and  Excited  Activity 510 

Theory  of  Radioactive  Changes 513 

Radioactive  Elements 515 

SOUND 

Nature  and  Propagation  of  Sound 517 

Musical  Sounds 526 

Sources  of  Musical  Sounds 536 

Velocity  of  Sound,  Experimental  Methods 544 

Acoustics  of  Halls 545 

References  and  Problems 550 

LIGHT 

General  Properties 553 

Velocity  of  Light 562 

The  Nature  of  Light 566 

Reflection 572 

Refraction  and  Dispersion 582 

Lenses    .    .    t 590 

Refraction  Phenomena 599 

Interference 604 

Diffraction 609 

Optical  Instruments  and  Measurements 617 

Emission  of  Radiant  Energy 631 

Absorption  of  Radiant  Energy 640 

Effects  Due  to  Absorption 644 

Double  Refraction  and  Polarization 649 

Dispersion  and  Selective  Reflection 667 

References  and  Problems 669 

List  of  Tables xv 

Greek  Letters  used  as  Symbols xvi 

INDEX  TO  NAMES 679 

INDEX  TO  SUBJECTS.                                                                                  .  683 


LIST  OF  TABLES 


MECHANICS  AND  PROPERTIES  OP  MATTER 

PAO» 

Coefficients  of  Diffusion 146 

Coefficients  of  Viscosity 132 

Compressibilities  of  Liquids 133 

Densities .    .  109 

Dimensions  of  Mechanical  Units 105 

Moduli  of  Elasticity 116 

Moments  of  Inertia 68 

HEAT 

Boiling  Points 249,  262 

Change  of  Boiling  Point  of  Water  with  Pressure 249 

Coefficients  of  Expansion  and  Pressure  of  Gases 223 

Coefficients  of  Expansion  of  Liquids 220 

Coefficients  of  Linear  Expansion 216 

Corrections  of  Gas  Thermometer 292 

Critical  Data 257 

Efficiencies  of  Steam  Engines 298 

Heats  of  Combustion 240 

Heats  of  Fusion .....  245 

Heats  of  Vaporization 252 

Melting  Points 242 

Specific  Heats  of  Gases  and  Vapors - 234 

Specific  Heats  of  Liquids  and  Solids 230 

Standard  Temperatures . 212 

Temperatures  on  Centigrade  Hydrogen  Scale 204 

Thermal  Conductivities 268 

Vapor  Tension  and  Vapor  Density  of  Water 246 

ELECTRICITY  AND  MAGNETISM 

Dielectric  Constants 364 

Dimensions  of  Electrical  and  Magnetic  Units 481 

Electrochemical  Equivalents 410 

Permeabilities  of  Iron 431 

Specific  Resistances 394 

Susceptibilities 434 

Units,  Electrical  and  Magnetic 480 

xiii 


xiv  LIST  OF  TABLES 

RADIOACTIVITY 

PAQB 

Radioactive  Elements 518 

SOUND 

Velocities  of  Sound 521 

Musical  Intervals 531 

Coefficients  of  Absorption 549 

LIGHT 

Fraunhof er  Lines 642 

Indices  of  Refraction  and  Dispersive  Powers 588,  589 

Indices  of  Refraction  of  Crystals 657 

Specific  Rotatory  Powers 665 

Wave  Lengths  in  General 640 

Trigonometric 675 

Sines  and  Cosines 677 

GREEK  LETTERS  USED  AS  SYMBOLS 

«  Alpha  e  Theta  p  Rho 

ft  Beta  K  Kappa  r  Tau 

7  Gamma  X  Lambda  4>  Phi 

a  Delta  /*  Mu  w  Omega 

17  Eta  TT  Pi 


TEXT-BOOK  OF  PHYSICS 


MECHANICS  AND  THE  PROPERTIES  OF 
MATTER 

BY  A.  WILMER  DUFF,  D.  Sc. 
Professor  of  Physics  in  the  Worcester  Polytechnic  Institute,  Worcester,  Mass. 

INTRODUCTION 

1.  Physics  as  a  Science. — From  the  evidence  of  our  senses  we 
infer  the  existence  of  a  great  variety  of  bodies  in  the  physical 
universe  around  us.  By  the  use  of  our  senses  we  also  learn  that 
these  bodies  have  various  characteristics  in  common,  such  as 
inertia,  weight,  and  elasticity,  and  these  we  attribute  to  the 
matter  of  which  in  various  forms  all  bodies  seem  to  consist. 
Matter  in  itself  is  inert;  the  mutual  actions  of  bodies  and  the  effects 
which  they  produce  on  our  senses  are  due  to  the  presence  in  them 
of  something  which  is  not  matter  and  which  is  called  energy. 
We  shall  define  the  word  energy  later;  the  thing  denoted  by  it  is 
known  to  all  as  the  means  which  are  supplied  by  the  sun,  fuels, 
and  elevated  bodies  of  water,  and  which  are  required  for  various 
familiar  operations  in  nature  and  industry. 

Physics  is  the  Science  of  the  Properties  of  Matter  and  Energy. — 
This  general  description  of  Physics  does  not  sharply  distinguish 
it  from  Chemistry  and,  in  fact,  no  definite  dividing  line  can  be 
drawn  between  the  two  sciences,  although,  in  a  general  way,  it 
may  be  stated  that  chemistry  deals  chiefly  with  questions  re- 
garding the  composition  and  decomposition  of  substances.  The 
different  branches  of  Engineering  also  treat  of  the  properties  of 
matter,  but  from  the  point  of  view  of  their  useful  applications. 

A  science  is  more  than  a  large  amount  of  imformation  on  some 
subject.  In  very  early  times  men  must  have  had  much  valuable 

1 


MECHANICS  ANIr  XE  PROPERTIES  OF  MATTER 


information  Tegar&ng'  the  physical  results  of  various  actions  and 
processes;  but  it  was  only  when  attempts  were  made  to  systema- 
tize and  arrange  this  knowledge  and  to  seek  the  relations  between 
the  different  facts  that  the  science  of  Physics  began.  The  descrip- 
tion of  the  phenomena  of  the  physical  world  became  more  and 
more  scientific  as  more  numerous  connections  between  physical 
phenomena  were  discovered  and  described.  At  the  present  time 
Physics  has  progressed  farther  in  this  direction  than  any  other 
science,  and,  in  seeking  to  give  a  brief  account  of  the  present  state 
of  the  science  of  Physics,  it  must  be  our  aim,  not  only  to  state  the 
most  important  observed  facts,  but  also  to  show  the  relations  and 
interdependence  of  these  facts. 

It  will  be  seen  as  we  proceed  that  in  some  parts  of  the  subject 
the  relations  between  observed  facts  are  better  understood  than 
in  other  parts.  Thus  in  Mechanics  the  relations  between  phe- 
nomena have  been  so  well  ascertained  that  we  are  able  to  start 
from  a  few  simple  laws  regarding  the  motions  of  bodies  and  from 
these  deduce  explanations  of  the  most  complicated  motions.  In 
other  parts  of  the  subject  we  must  be  content  to  take  from  time 
to  time  some  one  principle  and  trace  the  logical  consequences  of 
it  as  far  as  we  can,  and  then  proceed  to  do  the  same  with  other 
principles. 

After  classifying  and  studying  a  group  of  facts,  the  process  by 
which  we  arrive  at  some  underlying  principle  is  called  Induction. 
Thus,  the  principle  of  gravitation  was  discovered  by  Newton 
after  a  careful  comparison  of  the  motions  of  falling  bodies  and  of 
the  moon  and  the  planets.  Having  found  a  general  principle 
underlying  and  binding  together  many  phenomena,  we  may 
reason  forward  from  it  and  deduce  other  known  or  unknown  facts, 
as  in  Geometry  we  deduce  one  proposition  from  another.  This 
process  is  called  Deduction.  In  a  brief  account  of  Physics  we 
must  necessarily  use  deductive  more  frequently  than  inductive 
methods;  but,  where  space  will  permit,  the  effort  may  be  made 
to  show  how  by  induction  important  fundamental  principles 
have  been  discovered. 

2.  Measurement.  —  The  first  condition  for  success  in  tracing  the 
connection  between  the  facts  in  any  science  is  that  these  facts 
shall  be  ascertained  as  accurately  as  possible.  A  qualitative 
statement  of  the  size  or  weight  of  a  body,  to  the  effect  that  it  is 


INTRODUCTION  3 

large  or  small,  is  of  very  little  use.  A  quantitative  description 
of  the  same  consists  in  giving  the  ratio  of  its  size  or  weight  to 
that  of  some  accepted  standard.  Such  a  standard  is  called  a 
unit,  and  the  numerical  ratio  of  the  thing  measured  to  the  unit 
is  called  the  numerical  measure  (or  numeric)  of  the  thing 
measured. 

Some  measurements  are  direct,  that  is,  they  are  made  by  com- 
paring the  quantity  to  be  measured  directly  with  the  unit  of  that 
kind,  as  when  we  find  the  length  of  a  rod  by  placing  a  yard  or 
meter  scale  beside  it.  But  most  measurements  are  indirect. 
For  example,  to  measure  the  velocity  of  a  train  we  measure  the 
distance  it  travels  and  the  time  required,  and  by  calculation  we 
find  the  number  of  units  of  velocity  in  the  velocity  of  the  train. 

3.  Observation  and  Experiment. — In  some  branches  of  science 
mere  observation,  that  is,  taking  note  of  circumstances  and  events, 
is  the  chief  or  only  way  of  obtaining  knowledge.    For  example, 
the  astronomer  cannot  modify  the  motions  of  the  heavenly 
bodies;  he  must  be  content  to  observe.     Observation  also  plays 
an  important  part  in  Physics,  but  experiment,  which  consists  in 
modifying  circumstances  or  events  with  a  view  to  making  more 
valuable  observations,  plays  a  more  important  part.     Thus,  if 
we  desire  to  know  how  the  earth  attracts  a  body  and  whether  the 
attraction  is  different  at  different  places,  we  cannot  make  much 
progress  if  we  must  confine  ourselves  to  observing  bodies  falling 
freely  from  various  heights;  but,  if  we  modify  the  fall  by  attaching 
the  body  to  a  cord  and  swinging  it  as  a  pendulum,  we  are  able  to 
make  much  more  accurate  observations,  and  to  arrive  at  valuable 
information  that  we  could  probably  never  gain  by  observing  free 
falling  bodies.    For  this  reason  Physics  is  chiefly  an  experimental 
science,  that  is  to  say,  the  physicist  relies  on  carefully  planned 
experiments  to  find  information  and  then,  by  methods  of  reason- 
ing, and  especially  the  condensed  accurate  form  of  reasoning 
called  Mathematics,  he  extracts  from  the  results  of  the  experiment 
all  the  information  possible. 

4.  Hypotheses. — An  event  or  phenomenon  remains  obscure  or 
unexplained  when  its  logical  connection  with  other  events  or 
phenomena  has  not  been  traced.     But  it  is  explained  when  it  is 
shown  to  be  connected  with  other  familiar  phenomena,  and  the 
nature  of  the  connection   is  made  clear.     Thus,   the  rising  of 


4  MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

mercury  in  an  exhausted  tube  was  obscure  and  unexplained  until 
it  was  found  to  be  different  at  different  heights  along  a  mountain 
side  and  to  be  connected  with  the  pressure  of  the  air  on  the 
mercury  in  the  pool  in  which  the  tube  stands.  The  explanation 
in  such  a  case  consists  in  tracing  out  the  relation  of  cause  and 
effect  between  the  thing  explained  and  other  things.  The  latter 
may  themselves  be  still  unexplained.  Thus  the  way  in  which 
air  exercises  pressure  has  only  comparatively  recently  been 
explained. 

A  suggested  explanation,  while  its  correctness  is  still  in  doubt, 
is  called  an  hypothesis.  The  hypothesis  suggested  to  account  for 
the  pressure  of  air  (or  any  gas)  is  that  air  consists  of  flying 
particles,  which,  by  their  bombardment  of  a  surface,  produce 
what  we  call  the  pressure  on  the  surface;  this  suggested  explana- 
tion is  called  the  kinetic  hypothesis  of  gases.  The  formation  of 
an  hypothesis  plays  a  very  important  part  in  science,  for  it 
stimulates  research  to  test  its  truth;  and,  even  if  this  particular 
hypothesis  turn  out  inadequate,  in  testing  it  many  new  facts  are 
usually  ascertained  and  the  way.  is  paved  for  arriving  at  the 
right  explanation.  The  word  theory  is  sometimes  used  in  the 
same  sense  as  hypothesis,  but  it  is  better  to  restrict  it  to  meaning 
the  extended  discussion  of  an  explanation  or  verified  hypothesis. 
We  shall  use  it  in  this  sense  later  when  speaking  of  the  Kinetic 
Theory  of  Gases  (§227). 

5.  Cause  and  Effect.1 — When  a  certain  event  seems  inevitably 
to  be  followed  by  a  certain  other  event  we  are  accustomed,  in 
ordinary  language,  to  speak  of  the  former  as  the  cause  of  the 
latter,  and  of  the  latter  as  the  effect  of  the  former.  Thus,  the 
explosion  of  powder  in  a  gun  is  spoken  of  as  the  cause  of  the 
projection  of  the  bullet,  and  the  latter  event  is  described  as  the 
effect  of  the  explosion.  In  speaking  of  the  relation  of  two  things 
as  that  of  cause  and  effect,  we  do  not  merely  mean  that  one  has 
always  been  observed  to  follow  the  other,  but  we  suppose  that 
there  is  something  invariable  in  the  connection  between  them, 
that  is,  we  imply  our  belief  that  nature  will  always  act  in  the 
same  way  when  the  circumstances  are  the  same.  The  principle 

1  There  is  here  no  attempt  to  use  terms  in  a  critical  philosophical  sense.  The  use  of 
such  words  cannot  be  avoided  in  an  elementary  work  without  confusing  circumlocution 
and  they  must  be  used  here  in  tlieir  ordinary  sense. 


MECHANICS  5 

thus  stated  is  often  called  that  of  the  Uniformity  of  Nature. 
There  are,  however,  two  circumstances  which  must  be  considered 
as  of  no  importance  as  regards  the  connection  between  causes 
and  effects.  These  are  time  and  place.  The  time  of  an  event  is, 
of  course,  never  repeated,  and  nothing,  so  far  as  we  know,  ever 
comes  again  to  exactly  the  same  place,  since  the  sun  and  all  the 
planets  are  moving  rapidly  through  space. 

6.  Physical  Laws. — A  careful  study  of  any  phenomenon  usually 
enables  us  to  state  in  a  general  way  what  will  happen  in  certain 
circumstances.     Very  ancient  observation  led  to  the  conclusion 
that  bodies  when  unsupported  fall  toward  the  earth.     Such  a 
generalization  is  a  physical  law.     A  still  wider  study  usually  leads 
to  a  more  general  law.     Thus,  the  study  of  falling  bodies  and  of 
the  motion  of  the  moon  and  of  the  planets  led  Newton  to  the  con- 
clusion that  each  of  two  bodies  is  attracted  toward  the  other. 
The  aim  of  physical  research  is  to  obtain  physical  laws  of  increas- 
ing width  and  generality.     Any  such  law  is  very  imperfect  until 
it  can  be  stated  in  exact  mathematical  form,  and  this  requires 
careful  measurement.     By  measurement  and  calculation  Newton 
arrived  at  the  law  of  attraction  between  bodies  called  the  Law  of 
Universal  Gravitation.     Thus  a  physical  law  is  simply  a  state- 
ment that,  given  a  certain  set  of  circumstances,  certain  events 
will  follow  or  it  is  a  statement  of  some  aspect  of  the  Uniformity 
of  Nature. 

7.  Subdivisions  of  Physics.     The  Science  of  Physics  may,  for 
convenience,  be  divided  into  the  following  parts: 

1.  Mechanics.        3.  Heat.  6.  Sound. 

2.  Wave  Motion.  4.  Electricity  and  Magnetism.     7.  Light. 

5.  Radioactivity. 

The  subject-matter  of  each  of  these  parts  will  be  described 
when  that  part  is  taken  up. 

MECHANICS 

8.  Mechanics  is  the  branch  of  Physics  which  treats  of  the 
motions  of  bodies  and  the  causes  of  changes  in  these  motions.     It 
is  divided  into  two  parts,  one,  called  Kinematics,  in  which  the 
various  kinds  of  motion  are  described  and  studied,  and  the  other, 


6  MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

called  Dynamics,  in  which  the  .causes  of  change  of  motion  are 
studied.  Kinematics,  or  the  study  of  motion,  differs  from  Geom- 
etry in  having  to  consider  the  element  of  time.  Dynamics  is 
usually  divided  into  two  parts,  Kinetics  and  Statics,  the  former 
dealing  with  bodies  in  motion  and  the  latter  with  bodies  which, 
though  acted  on  by  causes  that  tend  to  produce  motion,  remain 
at  rest,  owing  to  the  fact  that  these  influences  counteract  each 
other.  (Some  authors  use  the  term  Dynamics  in  the  sense  here 
assigned  to  Kinetics.)  In  the  following  elementary  treatment 
of  Mechanics  it  will  not  be  convenient  to  treat  the  various  parts 
of  the  subject  quite  separately;  each  will  be  taken  up  in  turn 
as  convenience  and  simplicity  may  seem  to  dictate. 

KINEMATICS 
The  Geometry  of  Displacements 

9.  Translation  and  Rotation. — Motions  may  be  divided  into 
two  kinds.  A  moving  body  has  a  motion  of  translation  when 
every  straight  line  in  the  body  remains  parallel  to  its  original 
position.  .  Thus,  a  train  moving  on  a  straight  track  and  a  sled 
moving  down  a  uniform  incline  have  motions  of  translation.  In 
such  a  case  all  points  in  the  body  move  in  exactly  the  same  way. 
Hence  the  motion  of  the  body  is  completely  described  when  the 
motion  of  any  point  in  the  body  is  given,  and  we  may,  therefore, 
in  describing  the  motion  of  the  body,  treat  it  as  a  single  particle 
located  at  a  point. 

A  body  has  a  motion  of  rotation  when  all  points  in  the  body 
travel  in  circles  the  centers  of  which  lie  in  a  straight  line;  the 
line  is  called  the  axis  of  rotation.  This  is  the  motion  of  a  grind- 
stone, a  flywheel,  or  a  swing.  Any  two  points  in  such  a  body 
are  at  any  moment  moving  differently  (unless  they  lie  in  a  plane 
through  the  axis  and  are  equidistant  from  the  axis);  points 
farther  from  the  axis  move  in  larger  circles  and  more  rapidly 
than  those  nearer  to  the  axis. 

Many  forms  of  motion  are  highly  complex,  but  they  may  in  all 
cases  be  considered  as  made  up  of  translations  and  rotations. 

Since  the  motion  of  a  body  which  has  translation  without 
rotation  is  the  same  as  that  of  a  point,  it  is  convenient  to  begin 
with  a  study  of  the  motion  of  a  point. 


KINEMATICS  7 

10.  Position  of  a  Point. — The  position  of  a  point  is  fixed  by  its 
distances,  or  distances  and  directions,  from  other  points,  lines,  or 
surfaces.  The  simplest  way  of  stating  the  position  of  a  point 
is  by  giving  its  distance  and  direction  from  some  other  point 
which  we  may  call  the  starting-point  or  origin. 

When  we  confine  our  attention  to  points  in  a  certain  line, 
straight  or  curved,  their  positions  may  be  assigned  by  giving  the 
distance  of  each  point  from  some  assumed  origin  in  that  line. 
One  direction  away  from  the  origin  is  taken  as  positive  and  the 
opposite  direction  as  negative.  For  example,  the  position  of  any 
station  on  a  railway  line  may  be  fixed  by  its  distance,  positive  or 
negative,  from  some  other  station  taken  as  origin. 

When  we  confine  our  attention  to  points  on  a  surface,  plane 
or  curved,  the  position  of  each  point  may  be  assigned  by  its  dis- 
tance and  direction  from  some  origin  on  the  surface,  or,  what 
comes  to  the  same  thing,  by  its  distance  from  each  of  two  lines 
at  right  angles  passing  through  the  origin.  For  example,  a  point 
on  the  surface  of  the  earth  is  described  as  being  a  certain  distance 
east  or  west  and  a  certain  distance  north  or  south  from  the  origin. 

For  points  not  confined  to  any  line  or  surface  the  position  of 
each  may  be  assigned  by  its  distance  and  direction  from  some 
assumed  origin  in  space,  or,  what  comes  to  the  same  thing,  its 
distances,  positive  or  negative,  from  each  of  three  planes  inter- 
secting at  right  angles  at  the  origin. 

In  the  first  case  position  is  assigned  by  one  number,  in  the 
second  by  two  and  in  the  third  by  three.  A  point  is  said  to  have 
one  degree  of  freedom  when  its  motion  is  confined  to  a  definite 
line,  two  degrees  of  freedom  when  it  is  confined  to  a  definite  sur- 
face and  three  degrees  of  freedom  when  it  is  not  restricted  in  any 
way. 

The  above  statements  of  position  are  statements  of  relative  posi- 
tion, that  is,  statements  of  the  relation  of  the  position  of  a  point 
to  that  of  some  other  point  taken  as  origin.  Absolute  position, 
or  the  position  of  a  point  without  any  reference,  stated  or  im- 
plied, to  any  other  point  or  framework  of  lines,  could  not  be 
described  and  no  definite  meaning  could  be  attached  to  it.  In 
what  follows  the  word  position  will  always  mean  relative  posi- 
tion, and,  unless  otherwise  stated  or  implied,  the  point  of  refer- 
ence will  be  som«  point  on  the  surface  of  the  earth. 


8  MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

11.  Displacements. — A  change  of  position  is  called  a  displace- 
ment.    In  describing  a  displacement  we  do  not  need  to  make 
any  reference  to  the  time  in  which  the  point  moves  from  one 

position  to  the  other.  A  description  of  a  displace- 
ment  consists  in  a  statement  of  the  length  and 
direction  of  the  straight  line  drawn  from  the  first 
position  of  the  point  to  its  second  position.  Thus, 
i.— A  when  a  point  has  moved  from  A  to  B,  it  has  re- 
ceived  a  displacement,  the  magnitude  of  which  is 
the  lenSth  of  the  straight  line  AB  and  the  direc- 
tion of  which  is  the  direction  of  A  B.  This  dis- 
placement we  may  denote  by  the  symbol  AB  or  AB,  the 
arrow  or  stroke  being  placed  above  A  B  to  indicate  that  we  are 
referring  not  merely  to  the  length  of  the  line  A  B,  but  also  to 
its  direction  from  A  to  B. 

12.  Units  of  Length.     To  measure  or  specify  a  displacement  we 
must  use  some  unit  of  length.     The  unit  chiefly  employed  in 
Physics  is  the  meter  or  one  of  its  multiples  or  submultiples.     The 
meter  is  defined  as  the  distance  between  two  lines  on  a  bar  of 
platinum-iridium  kept  at  the  International  Bureau  of  Weights 
and  Measures  near  Paris,  when  the  temperature  of  the  bar  is  that 
of  melting  ice.     It  was  intended  by  the  designers  that  this 
length  should  be  one  ten-millionth  of  the  distance  from  a  pole 
of  the  earth  to  the  equator.     One  one-hundredth  of  the  meter 
is  called  the  centimeter  (0.01  m.),  and  this  is  the  unit  of  length 
which  we  shall  usually  employ.     Other  decimal  fractions  of  the 
meter  are  the  decimeter  (0.1  m.)  and  the  millimeter  (0.001  m.). 
For  great  distances  the  kilometer  (1000  m.)  is  employed. 

The  unit  of  length  popularly  used  in  English-speaking  countries 
is  the  yard  or  one  of  its  well-known  multiples  or  submultiples. 
The  British  yard  is  defined  legally  as  the  distance  between  two 
lines  on  a  bronze  bar  kept  at  the  office  of  the  Exchequer  in  Lon- 
don. The  legal  definition  of  the  yard  in  the  United  States  is 
£JH$  of  a  meter  (see  Vol.  I  of  the  Bulletin  of  the  Bureau  of  Stand- 
ards, Washington,  D.  C.). 

13.  The  Addition  of  Displacements. — If  the  point  that  moved 
from  A  to  B  did  not  travel  by  the  straight  line  A  B  but  passed 
through  points  C  and  D,  its  final  displacement  was  the  same  as  if 


KINEMATICS 


9 


it  had  gone  by  the  straight  line  AB;  but  the  final  displacement 
was  the  sum  of  a  number  of  separate  displacements,  AC, 
ClT,DB~.  Thus  ZB  is  the  resultant  or  sum  of  AC,  CD,  DB,  or  we 
may  say  that  by  adding  AC,  CD,  DB  we  get  A B,  or  briefly,  AB  — 
lAC  +  CD+DB;  but  it  must  be  carefully  noted  that  the  addition 
indicated  by  the  sign  4-  is  a  geometrical  process,  performed  by 
placing  the  displacements  end  to  end  as  the  sides  of  a  polygon 
and  taking  as  the  sum  the  displacement  from  the  initial  position 
to  the  final  position. 

If  from  C  we  draw  a  line  CD'  equal  and  parallel  to  DB,  and 
from  D'  a  line  D'B  equal  and  parallel  to  CD,  we  shall  have  another 
path  leading  from  A  to  B.  The  displacements  ~AC,  CD' ',  D'B 
added  together  give  the  same  sum  as  the  displacements  AC,  CD,DB 
added  together,  and  for  each  step  in  one  series  there  is  an  equal 
and  parallel  step  in  the  other  series.  It  is  evident  that,  so  far 
as  addition  of  displacements  is  concerned,  we  may  regard  CD'  and 
DB  as  the  same  displacement  and  D'B  and  CD  as  the  same  dis- 
placement. This  is  consistent  with  the 
definition  of  a  displacement  as  a  change  of 
position;  for,  when  a  point  goes  from  C  to  D, 
it  has  received  the  same  change  of  position 
as  another  point  has  received  when  it  has 
gone  from  D'  to  B,  CD  and  D'B  being  equal 
and  parallel.  Thus  all  displacements  which 
have  the  same  magnitude  and  direction  are 
equal. 

When  two  displacements  are  to  be  added,  the  addition  may  be 
performed  by  drawing  a  triangle.     Thus  to  add  A  B  and  BC  we 
complete  the  triangle  ABC  and  the  sum  is  AC. 
This  is  called  the  triangle  method  of  adding  two 
displacements.     Another  method  of  performing 
the   addition  is  to  construct  a  parallelogram. 
If  AD  be  drawn  from  A  equal  and  parallel  to 
BC,  the  displacement  AD  is  the  same  as  the 
displacement  BC  and  the  sum  of  AB  and  AD  is 
AC,  where  AC  is  the  diagonal  of  the  parallelogram  of  which  AB 
and  AD  are  adjacent  sides  drawn  away  from  A.     This  is  called 


Fio.  2. — Geometrical  ad- 
dition of  displacement. 


Fio.  3. 


10        MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

the  parallelogram  method  of  adding  two  displacements.  When 
several  displacements  are  to  be  added,  the  addition  is  performed 
by  constructing  a  polygon  as  in  Fig.  2. 

14.  Resolution  and  Subtraction  of  Displacements. — As  we  may 
replace  any  number  of  displacements  by  their  geometrical  sum  or 
resultant,  so  we  may  replace  a  displacement  by  any  number  of  dis- 
placements which  added  together  give  the  original  displacement. 
This  is  called  resolving  a  displacement  into  components.     Thus,  to 
resolve  a  displacement  AC  (Fig.  3)  into  two  components  in  given 
directions,  we  draw  from  A  lines  in  the  given  direction  and  then 
complete  the  parallelogram  ABCD  on  the  diagonal  AC',  Z§  and 
AD  are  the  components  desired,  since  their  sum  is  AC. 
Subtraction  is  the  opposite  of  addition.     To  subtract  4  from 
10  we  must  find  the  number,  6,  which  added  to 
4  will  give  10.     Similarly,  to  subtract  a  displace- 
merit,  PQ,  from  another,  PR,  we  must  find  the 
/     ^  displacement  which  added  to  PQ  will  give  PR. 

i-'"'  From  the  triangle  method  of  addition  this  is  evi- 

*\f  

dently  QR,  or,  if  we  complete  a  parallelogram 
traction  of  a  dis-  PQRS,  it  is  PS  which  is  equal  to  QR.  De- 
noting subtraction  by  the  minus  sign  PR  —  PQ 
=  QR=~PS. 

Subtraction  may  also  be  performed  in  a  slightly  different  way. 
From  the  triangle  method  it  is  evident  that  QP  added  to  PR 
will  give  QR.  Hence  to  subtract  a  displacement  we  may  reverse 
its  direction  and  add.  This  addition  may  also  be  performed  by 
a  parallelogram  PQ'SR  where  ~PQ'  =  QP.  Since  subtracting  ~PQ 
is  the  same  thing  as  adding  QP,  —PQ=+QP  or  the  minus 
sign  before  a  displacement  reverses  its  direction. 

15.  Vector  Quantities  and  Vector  Diagrams. — Displacements  be- 
long to  the  class  of  quantities  called  vector  quantities,  that  is,  quan- 
tities which  have  magnitude  and  direction.  Other  vector  quanti- 
ties are  velocities,  forces,  etc.  The  figures  in  the  preceding  sections 
are  diagrams  of  displacements,  that  is,  they  are  made  up  of  lines 
representing  the  actual  displacements  in  magnitude  and  direction. 
Thus  the  diagram  might  be  regarded  as  a  reduced  or  enlarged 


KINEMATICS  1 1 

picture  of  the  actual  displacements.  Other  vector  quantities, 
e.g.,  a  number  of  forces,  may  be  similarly  represented  by  a  vector 
diagram  by  drawing  lines  each  of  which  stands  in  magnitude  and 
direction  for  one  of  the  forces.  The  lines  in  such  a  diagram  are 
called  vectors.  The  lengths  of  any  two  vectors  in  such  a  diagram 
are  to  one  another  as  the  magnitudes  of  the  forces  represented, 
and  the  angle  between  the  two  vector.8  is  the  angle  between  these 
two  forces.  After  we  have  defined  the  meaning  of  the  resultant 
of  a  number  of  forces,  it  will  be  seen  that  it  is  represented  as  to 
magnitude  and  direction  by  the  vector  which  is  the  sum  of  the 
vectors  that  represent  the  separate  forces.  Similar  remarks 
apply  to  diagrams  of  velocities,  accelerations,  etc. 

Quantities  which  imply  no  reference  to  direction  are  called 
scalar  quantities.  Such  are  mass,  volume,  etc.  Each  such  quan- 
tity is  assigned  by  a  number  without  any  idea  of  direction  asso- 
ciated with  it,  and  the  addition  or  subtraction  of  such  quantities 
is  performed  in  the  ordinary  arithmetic  or  algebraic  manner. 

Velocity 

16.  Velocity  is  rate  of  change  of  position  or  rate  of  displace- 
ment.    Since  a  displacement  has  a  definite  direction  as  well  as  a 
definite  magnitude,  a  velocity  also  has  a  definite  direction  and  a 
definite  magnitude,   or  velocities  are    vector  quantities.     Thus 
"twenty  miles  an  hour"  is  not  a  complete  statement  of  a  velocity, 
since  it  gives  only  the  magnitude  of  the  velocity  and  does  not 
specify  its  direction;  but  "twenty  miles  an  hour  eastward"  is  a 
complete  statement  of  a  velocity.    For  clearness  such  a  phrase 
as  "twenty  miles  an  hour"  may  be  called  the  statement  of  a 
speed,   which   means   the  mere  magnitude  of  a  velocity  or  a 
rate  of  change  of  position  without  reference  to  the  direction 
of  the  change.     When  the  motions  considered  are  all  in  the 
same    straight  line  we  do  not  need  to    distinguish  speed   and 
velocity. 

17.  Constant  Velocity. — The  velocity  of  a  point  is  described  as 
constant  or  uniform  when  the  displacements  of  the  point  in  all 
equal  intervals  of  time  are  equal.     By  equal  displacements  must 
be  understood  displacements  equal  in  both  magnitude  and  direc- 
tion.    Hence,  when  the  velocity  of  a  point  is  constant,  the  point 


12          MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

moves  in  a  straight  line.  The  magnitude  of  a  constant  velocity 
is  measured  by  the  displacement  in  each  unit  of  time.  Hence, 
if  we  denote  the  magnitude  of  a  constant  velocity  by  v  and  the 
displacement  in  time  t  by  s, 

s  =  vt. 

Unit  velocity  is  the  velocity  of  a  point  that  travels  unit  distance 
in  unit  time,  e.g.,  1  cm.  in  1  sec. 

18.  Variable  Velocity. — A  point  has  a  variable  velocity  when  its 
displacements  in  equal  times  are  not  equal.   The  displacements  in 
successive  equal  intervals  of  time  may  differ  (1)  in  magnitude 
only,  as  when  a  point  moves  in  a  straight  line  with  varying  speed, 
or   (2)   in  direction  only,  as  when  a  point  moves  in  a  curve 
with  constant  speed,  or  (3)  in  both  magnitude  and  direction,  as 
when  a  point  moves  in  a  curve  with  varying  speed.     We  shall 
begin  by  considering  the  first  of  these  cases,  that  of  rectilinear 
motion. 

19.  Average  and  Instantaneous  Velocity. — In  rectilinear  motion 
with  variable  velocity  how  shall  we  define  the  magnitude  of  the 
velocity?     In  this  case  there  are  two  ways  open  to  us.     If  we 
divide  the  whole  distance  traversed  in  a  certain  interval  of  time 
by  the  length  of  the  interval  we  get  the  average  velocity  in  that 
interval.     If  for  example  we  find  the  whole  time  required  by  a 
train  to  move  from  one  station  to  another  on  a  straight  track  and 
divide  this  into  the  whole  distance,  we  get  the  average  velocity 
between  the  two  stations.   In  general,  denoting  the  whole  distance 
by  s,  the  whole  time  by  t,  and  the  average  velocity  by  v,  we  have 
v—s/t.     Hence 

s  =  vt. ' 

The  magnitude  of  the  average  velocity  in  an  interval  tells  us 
nothing  as  to  the  way  in  which  the  velocity  varies  during  the 
interval.  If  we  need  to  know  the  character  of  the  motion  more 
closely,  we  must  divide  the  whole  interval  into  parts  and  ascer- 
tain the  average  velocity  in  each.  The  smaller  these  parts,  the 
more  nearly  does  the  average  velocity  in  any  one  part  represent 
the  actual  velocity  at  any  moment  in  that  part.  Let  us  fix  our 
attention  on  a  certain  moment  at  a  time  t  after  the  beginning  of 
the  whole  interval.  If  we  proceeded  to  find  the  average  velocity 
in  a  short  interval,  say  A£,  including  that  moment,  and  if  we  took 


KINEMATICS  13 

successive  decreasing  values  for  At  and  found  the  average  velocity 
in  each  of  these  decreasing  values  of  At,  we  would  find  that  the 
average  velocity  would  rapidly  approach  a  definite  limiting  value. 
This  limiting  value  is  the  instantaneous  velocity  at  the  moment  t. 
Stated  more  briefly,  if  As  is  the  displacement  in  a  small  interval 
of  time  At  following  the  time  t,  the  instantaneous  velocity  at  the 
time  t  is  the  limiting  value  approached  by  As /At  as  At  approaches 
zero.  This  may  also  be  further  abbreviated  to  the  forms 


_  r As]  _  d 

[At\  At=0  =  ~d> 


ds} 

dt 


the  last  abbreviation  being  that  used  in  the  Differential  Calculus. 

When  the  velocity  of  a  point  is  constant,  the  instantaneous 
velocity,  as  defined  above,  is  the  same  as  the  velocity  of  the  point, 
as  defined  in  §  17.  For  the  values  of  As  /At  at  different  moments 
in  any  interval  t  are  equal.  Hence,  if  s  is  the  whole  distance  trav- 
ersed in  the  time  t,  each  value  of  As/ At  is  equal  to  s/t,  which  is 
the  distance  traversed  in  unit  time. 

When  the  instantaneous  velocity  of  a  point  is  variable,  we  may 
also  state  its  magnitude  in  terms  of  an  equal  constant  velocity. 
Suppose  that,  when  the  instantaneous  velocity  is  v,  the  point 
begins  to  move  with  a  constant  velocity  equal  to  v.  The  magni- 
tude of  this  constant  velocity  is  the  distance  the  point  would 
travel  in  unit  time.  Hence  we  derive  the  statement  that  the 
magnitude  of  the  instantaneous  velocity  of  a  point  is  equal  to  the 
distance  the  point  would  travel  in  unit  time  if  it  had  an  equal  con- 
stant velocity. 

20.  The  Unit  of  Time. — To  measure  or  specify  a  velocity  we 
must  use  some  unit  of  time.     The  unit  of  time  usually  employed 
in  Physics  is  the  mean  solar  second.     This  is  defined  as  -g-g-ihnr  °f 
the  mean  solar  day,  which  is  the  average,  thoughout  a  year,  of 
the  time  between  two  successive  transits  of  the  sun  across  the 
meridian  at  any  place.     It  is  the  second  of  the  ordinary  clock  or 
watch  when  it  is  properly  regulated. 

21.  Curvilinear  Motion. — When  the  displacements  of  a  point  in 
successive  equal  intervals  are  in  different  directions,  the  point  is 
moving  in  some  curved  path.     This,  for  example,  is  the  case  when 
a  ball  is  thrown  obliquely  upward  or  when  a  train  is  moving  on  a 
curved  track.     If  the  position  of  the  point  at  a  certain  time  t  IB 


14 


MECHANICS  AND  THE  PROPERTIES  OF  MATTER 


Fio.  5. 


P  and  at  a  somewhat  later  time,  say  (t  +  AO,  is  Q,  the  displace- 
ment in  this  time  is  PQ.  If  we  denote  the  length  of  PQ  byAs 
and  consider  the  limiting  value  of  As/ A*  as  before,  we  get  the 
instantaneous  velocity  of  the  point  at  the  time  t 
when  the  point  is  at  P.  As  PQ  is  decreased 
the  chord  PQ  finally  approaches  without  limit 
to  the  tangent  at  P;  hence  the  direction  of  the 
instantaneous  velocity  at  P  is  along  the  tangent 
at  P.  While  this  is  the  proper  meaning  of  the 
rate  of  displacement  at  P,  we  should  arrive  at 
the  same  value  for  the  instantaneous  velocity  if 
we  took  As  to  mean  the  length  of  the  arc  PQ, 
and  supposed  it  successively  diminished  by  the 
approach  of  Q  toward  P;  for  the  chord  and  the  arc  would  in 
the  limit  have  a  ratio  of  unity. 

22.  The  Graph  of  the  Speed  of  a  Point.— When  any  quantity  is 
variable,  much  valuable  information  can  frequently  be  derived 
from  the  properties  of  a  curve  drawn  to  represent  the  varying 
quantity.  A  curve  drawn  to  represent  the  speed  of  a  moving 
point  is  called  a  speed  curve.  Let  OA  be  a  straight  line  of  which 
the  length  OA  stands  for  the  length  of  the  interval,  t,  in  which 
we  wish  to  consider  the  motion.  Divide  OA  up  into  a  very  large 
number  of  small  equal  parts.  At  0  erect  a  perpendicular  OB  to 
represent  the  speed  at  the  beginning  of  the  interval  t.  Erect 
similar  perpendiculars  to  represent  the  instantaneous  values  of 
the  speed  at  the  beginnings  of  the  other 
parts  of  the  interval,  and  through  the  upper 
ends  of  these  perpendiculars  draw  a  smooth 
curve  BC. 

Consider  one  of  these  short  intervals,  ab. 
If  the  speed  throughout  this  short  interval 
had  been  the  same  as  at  the  beginning  of  the 
short  interval,  say  v,  the  distance  traversed 
in  the  short  interval  would  have  been  vXab 
or  the  unshaded  rectangle  above  db.    If  the 
speed  throughout  the  short  interval  had  been   the   same    as 
that  at  the  end  of  the  short  interval,  say  v',  the  distance 
would  have  been  v'Xab  or  the  area  of  the  unshaded  rectangle 
plus  that  of  the  small  shaded  rectangle  above  it.     The  real 


0          a   b  A 

Fid.  6. — Graph  of  a 
speed. 


KINEMATICS  15 

distance  in  the  interval  is  intermediate  between  these  two. 
Applying  the  same  reasoning  to  all  the  small  intervals  in  succes- 
sion, we  see  that  the  whole  distance  is  something  between  that 
represented  by  the  whole  unshaded  area  between  BC  and  OA 
and  that  represented  by  the  unshaded  area  plus  the  shaded  area. 
If  the  number  of  parts  into  which  OA  is  divided  be  doubled,  there 
will  be  twice  as  many  small  shaded  rectangles,  but  the  area  of 
each  will  only  be  one-fourth  as  great  as  before.  Hence  if  we  sup- 
pose the  number  of  the  small  intervals  increased  without  limit,  the 
shaded  area  will  decrease  without  limit  until  it  vanishes  and  the 
area  between  the  curve  BC  and  the  line  OA  will'  represent  the 
actual  distance  in  the  time  t. 

Since  it  is  merely  the  magnitude  of  the  velocity  that  is  repre- 
sented by  each  ordinate,  the  area  represents  the  distance  measured 
along  the  line  of  motion,  whether  this  be  straight  or  curved.  Thus, 
if  a  point  moves  once  around  a  circle  with  constant  speed,  BC  will 
be  a  horizontal  straight  line,  and  the  distance  represented  by  the 
area  OBCA  will  equal  the  circumference;  but  the  mean  velocity  in 
the  revolution  will  be  zero,  since  the  final  displacement  will  be  zero. 

To  bring  out  more  clearly  the  meaning  to  be  attached  to  the 
word  "represent"  in  the  above,  let  us  first  suppose  that  OA 
contains  as  many  units  of  length  as  t  contains  units  of  time,  and 
that  OB  contains  as  many  units  of  length  as  the  velocity  it  stands 
for  contains  units  of  velocity.  Each  unit  of  area  will  then 
stand  for  a  unit  of  distance  traversed  by  the  moving  point,  and 
the  whole  area  will  contain  as  many  units  of  area  as  the  distance 
traversed  contains  units  of  length.  But  if  each  unit  of  length 
along  OA  stands  for  m  units  of  time  and  each  unit  of  length 
along  OB  stands  for  n  units  of  velocity,  the  whole  area  will  be  mn 
times  smaller  than  it  would  have  been  on  the  first  supposition, 
and,  to  get  the*  whole  distance,  we  shall  have  to  multiply  the 
whole  area  by  mn. 

23.  The  Resultant  of  Simultaneous  Velocities. — A  man  sitting 
in  a  train  has  the  velocity  of  the  train,  but,  when  he  gets  up  and 
moves  about,  he  has  an  additional  velocity  which  may  or  may  not 
be  in  the  same  direction  as  the  first  velocity.  Similarly  a  launch 
floating  down  with  the  current  in  a  river  has  the  velocity  of  the 
current;  but  if  it  has  a  propeller  in  motion,  it  has  another  velocity 
in  addition  to  the  first.  When  a  body  has  two  or  more  simul- 


16          MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

taneous  velocities,  it  pursues  some  definite  path  and  its  velocity 
in  the  path  is  called  the  resultant  of  the  simultaneous  velocities. 

From  this  definition  of  the  resultant  of  any  number  of  simul- 
taneous velocities  it  can  be  shown  that  the  magnitude  and  direc- 
tion of  the  resultant  velocity  can  be  deduced  from  the  separate 
velocities  by  the  triangle,  parallelogram,  or  polygon  method  of 
adding  vectors.  Consider  first  the  case  of  two  constant  velocities 
and  draw  a  diagram,  in  which  A B  and  AC  stand  for  the  two 
velocities-  Complete  the  parallelogram  ABCD.  We  shall  show 
that  AD  stands  for  the  resultant  velocity.  Since  the  velocities 
are  constant  AB  and  AC  represent  in  magnitude  and  direction  the 
component  displacements  in  unit  time,  and  the 

• 

sum  of  these  displacements  is  represented  by  AD 
(§  13),  which,  therefore,  represents  the  resultant 
A  C          displacement  in  unit  time.     Hence  AD  represents 

FIO.  7.  the  resultant  velocity.     Thus  the  parallelogram 

method  applies  to  the  addition  of  constant 
velocities,  and  the  same  must  be  true  of  the  other  methods 
which  are  essentially  the  same. 

When  the  component  velocities  are  not  constant,  we  can  add 
their  instantaneous  values  by  the  vector  methods  referred  to.  The 
proof  of  this  statement  is  the  same  as  above,  except  that  AB,  AC, 
and  AD  will  now  stand  for  the  displacements  that  would  take 
place  in  unit  time  if  the  velocities  remained  constant  that  long. 
24.  Formula  for  Resultant. — Let  vt  and  v2  be  the  respective 
magnitudes  of  two  component  velocities  of  a  moving  point,  and 
let  these  velocities  be  represented  by  OA  and  OB  (Fig.  8). 
Also  let  v  be  the  magnitude  of  the  resultant  velocity,  which  is  rep- 
resented by  OCj  where  OC  is  the  diagonal  of  the  .parallelogram  of 
which  OA  and  OB  are  sides.  By  a  well-known  trigonometrical 
formula 

OC3  =  OAa  + AC2-2<L4-AC  cos  OAC 

Denote  the  angle  AOB,  which  is  the  angle  between  the  directions 
of  the  two  components,  by  0.  Then  the  angle  OAC  equals 
(180°- 0)  and  therefore  cos  OAC=  -cos  6.  Since  OA,  OB,  and 
OC  are  proportional  to  vlt  vt,  and  v  respectively, 

cos  6 


KINEMATICS 


17 


By  this  formula   we   can  calculate  the  magnitude  of  v  when 
i>1;  v2>  and  0  are  known. 

When  0  =  0,  that  is,  when  the  components  are  in  the  same  direc- 
tion, cos  0  =  1  and  the  formula  for  v  gives  v  =  (vl+v2).  When 
0  =  180°,  that  is,  when  the  components  are  in  opposite  directions, 
cos  0  =  —  1  and  v  =  ±  (v^  —  v- 


If  0  =  90°,  that  is,  if  the  components  are  at  right  angles,  cos  6  =  0 
and  (Fig.  9) 


and  if  $  be  used  to  denote  the  angle  AOC  which  the  resultant 
makes  with  the  component  of  magnitude  vl} 

AC    t>2 

tan  <£  =  ---  =  - 
OA     vl 

25.  Resolution  of  a  Velocity  into  Components.  —  Since  two  ve- 
locities taken  together  are  equivalent  to  a  single  velocity  called 
their  resultant,  we  may  reverse  the  process  and  suppose  any  ve- 
locity replaced  by  any  two  velocities  which  added  are  equivalent 
to  the  original  velocity.  This  is  called  resolving  a  velocity  into  com- 
ponents. To  thus  resolve  a  velocity  we  must  draw  a  parallelo- 
gram of  which  the  diagonal  stands  for  the  velocity  to  be  resolved. 
Now  any  number  of  parallelograms  can  be  drawn  with  a  given 
line  as  diagonal;  but,  if  the  directions  of  the  sides  are  specified, 
only  one  solution  is  possible.  Hence  to  resolve  a  given  velocity 
into  components  in  two  given  directions  is  a  definite  problem, 
which  may  be  solved  graphically  by  constructing  a  parallelogram. 

The  most  important  case  of  the  above  is  when  the  directions 
of  the  components  are  at  right  angles.  Thus,  if  the  velocity  is 
v  in  the  direction  Oc  and  if  OC  is  taken  to  represent  v  and  if  Oa 
and  Ob  are  to  be  the  directions  of  the  components,  we  draw 
from  C  perpendiculars,  CA  and  CB,  to  Oa  and  06  respectively. 


18          MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

Then  OB  and  OA  are  the  desired  components  in  the  specified 
directions.  If  the  direction  Oa  makes  an  angle  0  with  the 
direction  of  v  and  if  we  denote  the  components  in  the  directions 
Oa  and  Ob  respectively  by  v^  and  v2, 

vl  —  v  cos  0.    v2  =  v  sin  6. 

It  should  be  noted  that  0  stands  for  an  angle  that  may  be  either 
positive  or  negative.     We  may  regard  0  as  the  angle  through 
which  a  line,  starting  from  the  position  Oa,  must 
''     revolve  about  0  to  reach  the  position  Oc;  and 
when   the   revolution  is  counter-clockwise,  it  is 
customary  to  regard  such  an  angle  as  positive,  the 
1     opposite  direction  of  revolution  corresponding  to  a 
FIG.  10.         negative  angle.     If  we  make  this  agreement  as 
regards  the  sign  of  0,  we  must  keep  to  it  as  regards 
the  right  angle  that  Ob  makes  with  Oa,  that  is  to  say,  the  right 
angle  and  the  angle  6  must  be  measured  away  from  Oa  in  the 
same  direction,  namely,  counter-clockwise. 

Acceleration 

26.  Acceleration  is  rate  of  change  of  velocity.  A  change  of 
velocity  has  a  definite  direction  as  well  as  a  definite  magnitude. 
Hence  acceleration  is  a  quantity  which  has  both  direction  and 
magnitude,  that  is,  acceleration  is  a  vector  quantity. 

An  acceleration  may  be  either  constant  or  variable.  The  ac- 
celeration of  a  point  is  constant  when  the  velocity  of  the  point 
changes  by  equal  amounts  in  equal  intervals  of  time.  By  equal 
changes  of  velocity  must  be  understood  changes  of  velocity  that 
are  equal  in  magnitude  and  in  the  same  direction.  When  the 
changes  of  velocity  in  equal  intervals  of  time  are  not  equal,  the 
acceleration  is  variable. 

The  statement  that  the  velocity  of  a  point  is  variable  may 
refer  to  a  change  in  the  magnitude  of  the  velocity,  to  a  change  in 
the  direction  of  the  velocity,  or  to  a  change  in  both.  Hence  we 
shall  have  three  cases  of  acceleration  to  consider:  (1)  the  accelera- 
tion of  a  point  when  the  velocity  of  the  point  is  constant  in 
direction  but  variable  in  magnitude;  (2)  the  acceleration  of  a 
point  when  the  velocity  of  the  point  is  constant  in  magnitude 


KINEMATICS  19 

but  variable  in  direction;  (3)  the  acceleration  of  a  point  when 
the  velocity  of  the  point  is  variable  in  both  magnitude  and 
direction. 

The  simplest  case  is  when  the  velocity  of  the  moving  point 
is  constant  in  direction  and  when  the  acceleration  is  con- 
stant and  in  the  direction  of  the  line  of  motion.  This  is 
illustrated  by  a  body  dropped  from  a  height  and  falling  in  a 
straight  line. 

The  magnitude  of  a  constant  acceleration  is  the  magnitude  of 
the  velocity  added  in  each  unit  of  time,  and  the  direction  of  the 
acceleration  is  the  direction  of  the  added  velocity.  The  unit  of 
acceleration  is  that  of  a  point  the  velocity  of  which  increases  by 
unit  velocity  in  unit  time.  When  the  cm.  is  taken  as  unit  of 
length  and  the  sec.  as  unit  of  time,  the  unit  of  acceleration  is 
such  that  the  velocity  increases  by  one  cm.  per  sec.  in  each 
second,  or,  briefly,  one  cm.  per  sec.  per  sec. 

27.  Motion  in  a  Straight  Line  with  Constant  Acceleration. — In 
considering  the  motion  of  a  point  along  a  straight  line,  we  take 
one  direction  along  the  line  as  positive  and  the  opposite  direction 
as  negative  and  we  do  not  need  to  distinguish  between  speed  and 
velocity  (§16).  Let  t?0  be  the  velocity  of  the  point  at  the  begin- 
ning of  an  interval  of  time  of  length  t,  and  let  v  be  its  velocity  at 
the  end  of  the  interval.  The  increase  of  velocity  is  (v  —  VQ) 
and  the  increase  per  unit  time  is  (v  —  v0)/t.  This  is,  therefore, 
the  magnitude  of  the  constant  acceleration,  which  we  shall  denote 
by  a.  Hence 

v  =  v0  +  at  (1) 

This  very  important  equation  is  simply  a  statement  that  the 
final  velocity  (at  the  end  of  the  time  t)  is  equal  to  the  initial 
velocity  (at  the  beginning  of  t)  plus  the  increase  of  velocity,  and 
the  increase  of  velocity  is  equal  to  the  acceleration  multiplied 
by  the  time. 

To  find  how  far  the  point  travels  in  the  time  t  let  us  consider 
the  form  of  the  velocity  curve  (§22)  in  the  present  case.  The 
changes  of  velocity  in  equal  short  intervals  of  time  are  equal. 
Hence,  in  Fig.  6,  the  differences  between  each  ordinate  and  the 
next  in  order  are  equal,  and  the  velocity  curve  is  therefore  a 
straight  line,  as  in  Fig.  11.  Draw  BD  parallel  to  OA.  The 
whole  area  OBCA  consists  of  two  parts,  that  of  the  rectangle 


20  MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

0   DA  and  that  of  the  triangle  BDC.     OB  represents  the  initial 

v  ocity  VQ,  and  we  shall  suppose  that  the  figure  is  drawn  to 
such  a  scale,  that  OB  contains  as  many  units 
of  length  as  v0  contains  units  of  velocity,  and 
that  the  same  is  true  of  AC,  which  represents 
the  final  velocity  t>.  The  height  of  the  tri- 
angle, DC,  represents  in  the  same  way  the 
increase  of  velocity  at.  OA  represents  the 
time  t,  and  we  shall  suppose  that  OA  contains 
the  same  number  of  units  of  length  as  t  con- 

tains units  of    ime.     The  whole  area  is  therefore  (vQt  +  ^t-at). 

Hence,  if  s  is  the  whole  distance  traversed  in  the  time  t, 

s  =  v0t  +  $at*  (2) 

This  very  important  equation  consists  of  two  parts.  The  part, 
vQt,  is  the  distance  the  point  would  have  travelled  in  the  time  t, 
if  its  velocity  throughout  t  had  remained  constant  and  equal  to 
the  initial  velocity  VQ.  The  part  %at*  is  the  additional  distance 
due  to  the  acceleration,  that  is,  the  distance  the  point  would 
have  gone  if  it  had  started  from  rest  with  an  acceleration  a. 

Between  (1)  and  (2)  we  may  eliminate  t  and  so  find  an  expres- 
sion for  the  final  velocity  in  terms  of  the  initial  velocity,  the 
acceleration,  and  the  distance. 


(3) 

Equations  (1),  (2),  and  (3)  are  of  great  importance. 

Another  expression  for  the  area  OBCA  is  %(AC  +  OB)-OA. 
Hence  the  distance  is  also  given  by  the  formula 


From  this  it  follows  that  the  average  velocity,  which  equals  the 
total  distance  divided  by  the  time  (§19),  is  equal  to  one-half 
of  the  sum  of  the  initial  velocity  and  the  final  velocity. 

Equation  (2)  may  be  readily  obtained  by  means  of  the  Integral  Calculus. 
The  distance  travelled  in  a  short  time  dt  when  the   velocity  is  v  is  vdt. 

Hence  the  whole  distance,  s,~ 


KINEMATICS  21 

28.  Galileo's  Experiments.  —  The  very  important  relations  ex- 
pressed by  (1)  and  (2)  were  discovered  by  Galileo  by  studying 
the  motion  of  falling  bodies,  and  this  discovery  was  the  beginning 
of  Kinetics.     Before  that  time  nothing  was  known  as  to  the  way 
in  which  the  velocity  of  a  body  increases  as  it  falls.     Galileo 
thought  the  law  of  increase  expressed  by  (1),  namely,  that  the 
increase  of  velocity  is  proportional  to  the  time,  was  probably 
correct;  but  the  instrumental  means  at  his  command  did  not 
enable  him  to  test  it;  so  he  deduced  (2),  practically  by  the  graph- 
ical method  given  in  §27,  and  then  tested  it.     To  avoid  having 
to  deal  with  any  great  velocities,  such  as  that  of  a  body  falling 
vertically,  he  tested  the  rolling  of  a  ball  down  an  inclined  plane, 
assuming  that  both  motions  would  follow  the  same  law.     The 
result  confirmed  his  formula. 

29.  Acceleration  of  Free  Fall.  —  We  shall  assume  as  an  experi- 
mental fact,  discovered  by  Galileo,  that  at  any  one  place  all  bodies 

'  falling  freely  would  have  the  same  acceleration,  if  it  were  not  for 
the  effect  of  air  friction.  The  latter  is  very  small  in  the  case 
of  dense  solids,  such  as  blocks  of  metal,  falling  moderate  dis- 
tances, and  may  usually  be  neglected.  The  acceleration  of  free 
fall,  or  the  acceleration  of  gravity,  as  it  is  often  called,  is  usually 
denoted  by  g.  In  the  c.  g.  s.  system  g  is  about  980  cm.  per  sec. 
per  sec.,  though  slightly  different  at  different  points  on  the 
earth's  surface,  and  in  feet  and  seconds  it  is  about  32.2  ft.  per 
sec.  per  sec.  Hence,  from  §27,  when  a  body  is  projected  verti- 
cally downward  with  a  velocity  VQ,  its  velocity  and  distance 
after  an  interval  t  may  be  found  from 


When  the  direction  of  projection  is  upward  we  may  take 
upward  as  the  positive  direction,  and  g,  being  downward,  will 
then  be  negative.  In  this  case 

«  =  »o-0*,  (1) 

«  =  »o*-i#a,  (2) 

v*  =  v0*-2gs  (3) 

At  the  highest  point  i?  =  0;   hence  from  (1)    we  have  t=vQ/g. 
Substituting  this  in  (2),  we  get  for  the  height  of  ascent  s  =  $v02/g. 


22          MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

This  also  follows  from  (3)  by  putting  v  =  0.  The  time  of  return 
to  the  ground  is  got  by  putting  s  =  0  in  (2).  This  gives  t  =  2v<>lg, 
showing  that  the  whole  time  of  rise  and  fall  equals  twice  the  time 
of  ascent,  or  that  the  time  of  rise  equals  the  time  of  fall.  It  fol- 
lows from  (3)  that  the  velocity  of  return  to  the  starting  point, 
that  is,  when  s  is  again  zero,  equals  the  velocity  of  projection  in 
magnitude,  but  it  is  in  the  opposite  direction.  It  must,  however, 
be  remembered,  that  these  statements  are  true  only  for  moderate 
velocities.  At  high  velocities,  such  as  those  of  a  bullet,  air- 
resistance  greatly  modifies  the  motion. 

The  value  of  g  at  any  station  of  observation  depends  on  the  latitude  of  the 
station  and  also  on  the  height  of  the  station  above  sea  level.  The  results 
of  very  careful  experiments  show  that,  at  a  station  in  latitude  A  and  at  an 
elevation  of  I  meters  above  sea  level, 

0  =  977.989  (1  +  .0052  sin2;  -  .0000002  I) 

30.  Motion  of  a  Projectile. — When  a  body  is  thrown  obliquely 
into  the  air,  its  motion  may  be  considered  as  consisting  of  a 
horizontal  part  and  a  vertical  part.     The 
vertical  part  is  subject  to  a  constant  ac- 
celeration g  downward;  while,  since  there 


I 

x      is  no  horizontal  acceleration  (if  we  may 


FIO  12  —Path  of  a        neglect  air-friction)  ,  the  horizontal  part  of 

projectile.  the  motion  is  a  constant  velocity.     If  the 

magnitude  of  the  velocity  of  projection  is 

v  and  the  direction  of  projection  makes  an  angle  0  with  the 

horizontal,  the  velocity  may  be  resolved  into  a  component  v  cos  0 

in  a  horizontal  direction  and   a  component  v  sin  0  in  a  direc- 

tion vertically  upward.     If,  then,  x  is  the  horizontal  distance 

traversed  in  time  t, 

cos  6  (1) 


and  if,  at  the  time  t,  the  vertical  distance  attained  is  yt 

y=vtsm  0-tgt*  (2) 

Thus  the  vertical  motion  is  the  same  as  that  of  a  body  thrown 
vertically  upward  with  a  velocity  v  sin  6.  Hence  (§29)  at  time 
(v  sin  6)/g  the  body  will  have  just  lost  its  vertical  velocity  and 
will  therefore  be  moving  wholly  in  a  horizontal  direction;  and 
at  that  moment  the  height  will  be  (v3  sin2  0)/2g.  At  the  time 


KINEMATICS  23 

(2v  sin  0)/g  the  body  will  have  returned  to  its  original  level  and 
the  distance  horizontally  from  its  starting-point  will  then  be 
v  cos  6-(2v  sin  8)/g  or  (v2  sin  26} Ig.  Now,  since  sin  2e  has  its 
maximum  value,  unity,  when  20  is  90°,  that  is,  when  0  is  45°, 
it  follows  that  the  greatest  horizontal  range  for  a  given  velocity, 
v,  of  projection  is  v2/g  and  is  obtained  by  making  the  angle  of 
projection  45°. 

If  it  be  desired  to  find  the  constant  relation  that  holds  between  x  and  y 
during  the  motion,  the  value  of  t  taken  from  (1)  may  be  substituted  in  (2) 
and  we  shall  get 

y~x  tan  0-x20/2va  cosa0 

the  equation  of  a  parabola  referred  to  axes  through  the  point  of  projection. 
Hence  the  path  of  the  projectile  is  a  parabola. 

As  in  the  case  of  §29,  these  results  are  approximately  correct  only  in  the 
case  of  the  moderate  velocities  for  which  air-friction  is  negligible.  (See 
article  on  "Ballistics"  Ency.  Britt.,  llth  edition.) 

31.  Variable  Acceleration.— When  the  acceleration  of  a  point 
is  variable,  we  can  no  longer  measure  it  by  the  actual  increase  of 
velocity  in  any  time.     We  may,  however,  divide  the  magnitude 
of  the  increase  of  velocity  in  any  time  by  the  time  and  call  this 
the  magnitude  of  the  average  acceleration  in  that  time,  the  direc- 
tion of  this  average  acceleration  being  the  direction  of  the  in- 
crease of  velocity.     The  instantaneous  value  of  the  acceleration  is 
defined  much  as  in  the  case  of  instantaneous  velocity,  namely, 
as  the  value  to  which  the  average  acceleration  approaches  as  the 
interval  is  diminished  without  limit,  or 

^dv 
=o     dt 

A  variable  acceleration  may  be  variable  as  regards  magnitude 
or  direction  or  both.  In  the  following  we  shall  consider  the  case 
of  an  acceleration  that  is  constant  in  magnitude  but  variable  in 
direction. 

32.  Acceleration  of  a  Point  Which  Moves  in  a  Circle  with 
Constant  Speed. — Let  P  be  the  position  of  the  moving  point  at 
time  t  and  P'  its  position  at  time  t  4-  A£.     At  P  the  point  is  moving 
in  the  direction  of  the  tangent  PT  and  at  P'  in  the  direction  of 
the  tangent  P'T'  (Fig.  13). 

From  any  point  0  draw  lines  OQ,  OQ'  of  equal  length  to  repre- 
sent the  velocities  v  and  v'  at  P  and  P'  respectively.  QQ'  will 


24 


MECHANICS  AND  THE  PROPERTIES  OF  MATTER 


represent  the  velocity,  Av,  added  in  time  A£.     The  triangles 
OQQ'  and  OPP'  are  similar.     The  arc  PP'  equals  v&t  and  the 
chord  PP'  approaches  equality  with  the  arc  PP'  as  A£  is  di- 
minished.    Hence  approximately 

0  Av  _  vAZ 

v       r 

and    this   relation   becomes    more 
nearly  exact  as  A£  is  diminished. 


Hence  (§31) 


a  =  -- 


QQ'  is  perpendicular  to  PP'  and  in 
the  limit  it  is  in  the  direction  PC. 

Hence  the  acceleration  is  directed  toward  the  center.     It  is,  therefore, 
an  acceleration  of  constant  magnitude  but  of  variable  direction. 

33.  Curvilinear  Motion. — If  in  the  preceding  the  speed  were  not  constant, 
there  would,  in  addition  to  the  acceleration  toward  the  center,  be  an  acceler- 
ation along  the  tangent.    The  first  acceleration  would  have  the  effect  of 
changing  the  direction  of  the  velocity,  while  the  second  would  have  the 
effect  of  changing  the  magnitude  of  the  velocity,  that  is,  the  speed. 

When  a  point  moves  with  constant  speed  in  a  curve  of  any  form,  it  may 
be  regarded  as  moving  at  any  moment  in  a  circle  which  coincides  at  that 
point  with  the  curve;  this  circle  is  called  the  circle  6f  curvature  at  that  point 
on  the  curve.  From  the  radius  of  the  circle  of  curvature  at  a  point  on  the 
curve  we  can  calculate  the  acceleration  toward  the  center  of  the  circle  of 
curvature.  If  the  speed  of  the  point  is  not  constant,  the  point  must  also 
have  an  acceleration  along  the  tangent  to  the  curve. 

34.  Addition  of  Accelerations. — A  moving  point  may  have  two 
or  more  accelerations  simultaneously.     Thus,  a  man  at  rest  on 
the  deck  of  a  ship  which  is  moving  with  an  ac- 
celeration has  one  acceleration,  that  of  the  ship. 

If  he  moves  across  the  deck  with  an  acceleration 
independent  of  the  motion  of  the  ship,  he  has  a 
second  acceleration.  In  any  such  case  the  mov- 
ing body  travels  in  some  curve  with  a  definite 
acceleration  which  is  called  the  resultant  of  the  component 
accelerations. 

We  can  readily  show  that  the  resultant  acceleration  may  be 
deduced  from  the  component  accelerations  by  the  vector  method 


KINEMATICS  25 

of  addition,  that  is,  by  the  construction  of  a  triangle,  parallelo- 
gram or  polygon,  the  sides  of  which  represent  the  separate 
accelerations.  For  let  A  B  and  AC  represent  two  constant 
accelerations  possessed  simultaneously  by  a  point.  Since  the 
acceleration  represented  by  A  B  is  constant,  the  change  of  veloc- 
ity it  produces  in  unit  time  is  also  represented  by  A  B.  Simi- 
larly AC  represents  the  change  of  velocity  in  unit  time  due  to 
the  second  constant  acceleration.  The  resultant  change  of 
velocity  is  found  by  completing  the  parallelogram  ABDC; 
hence  AD  is  the  resultant  change  of  velocity  in  unit  time,  that  is, 
the  resultant  acceleration.  The  same  method  of  reasoning  is 
applicable  when  the  accelerations  are  variable;  the  only  differ- 
ence being  that  AB  and  AC  and  AD  all  represent  velocities  that 
would  have  been  added  in  unit  time,  if  the  accelerations  remained 
constant  that  long. 

35.  Resolution  of  an  Acceleration  into  Components.  —  Since  two 
or  more  accelerations  may  be  replaced  by  their  resultant,  it  fol- 
lows that  an  acceleration  may  be  resolved  into  two  or  more  com- 
ponents by  the  ordinary  methods.  The  case  in  which  an  accel- 
eration is  resolved  into  two  components  at  right  angles  is  espe- 
cially important.  As  an  example,  suppose  a  body  rests  on  a 
smooth  plane  the  inclination  of  which  to 
the  horizontal  is  i.  If  the  body  were  not 
supported,  it  would  fall  with  an  accelera- 
tion of  g.  The  acceleration  may  be  re- 
solved into  a  component  gcoai  perpen- 
dicular to  the  plane  and  a  component 
gsmi  parallel  to  the  plane.  The  com-  Fio  15._Acceleration  down 
ponent  perpendicular  to  the  plane  has  no  a  plane. 

effect,  since  motion  perpendicular  to  the 

plane  is  prevented,  whereas  the  other  component  causes  it  to 
slide  down  the  plane  with  an  acceleration  g  sin  i.  Thus  the 
motion  down  the  plane  may  be  calculated  by  the  formulae  of 
§29,  a  being  replaced  by  g  sin  i. 

From  this  we  may  deduce  one  result  of  importance.     The  ve- 
locity after  a  distance  of  descent  s  down  the  plane  is  given  by 


a  +  20  sin 


26          MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

and,  if  h  is  the  distance  of  descent  measured  vertically,  h  =  s  •  sini. 
Hence 


Now  this  is  the  formula  we  should  have  been  led  to,  if  we  had 
sought  the  velocity  attained  by  a  free  vertical  fall  through  a  dis- 
tance h.  Hence  the  speed  attained  by  a  body  which  slides  without 
friction  through  a  certain  vertical  distance  is  the  same  as  if  the 
body  had  fallen  that  distance  vertically.  This  does  not  apply  to 
a  body  rolling  down  a  rough  plane. 

DYNAMICS 
Force  and  Mass 

36.  In  the  preceding  we  have  considered  various  cases  of  mo- 
tion without  any  reference  to  the  influences  that  affect  the  motions 
of  bodies,  just  as  in  Geometry  we  study  lines  and  figures  without 
any  reference  to  particular  bodies.     We  must  now  consider  those 
relations  between  bodies  on  which  changes  of  motion  depend. 

Isaac  Newton  was  the  first  who  attained  clear  ideas  as  to  the 
relations  between  bodies  and  their  motions.  His  treatment  of  the 
subject  was  founded  on  three  fundamental  principles  which  he 
called  Axioms  or  Laws  of  Motion.  These  axioms  are  so  simple 
that  they  are  recognized  as  very  probably  true  as  soon  as  their 
meaning  is  grasped.  The  proof  of  their  correctness  is,  however, 
the  fact  that  all  deductions  from  them  are  found  to  be  verified  by 
observation  and  experiment. 

37.  Newton's  First  Law  of  Motion.  —  Every  body  persists  in  its 
state  of-  rest  or  of  uniform  motion  in  a  straight  line,  unless  it  is 
compelled  by  some  force  to  change  that  state. 

This  law  may  be  divided  into  two  parts,  a  statement  and  a  defi- 
nition. The  statement  is  that  any  change  of  velocity  of  a  body, 
that  is  any  acceleration,  is  due  to  some  external  influence,  and  a 
body  free  from  external  influences  would  necessarily  have  a  con- 
stant velocity.  This  was  a  complete  denial  of  what  had  been 
supposed  to  be  true  up  to  the  time  of  Galileo  (who  died  in  1642, 
the  year  in  which  Newton  was  born)  ;  for,  until  then,  it  was  sup- 
posed that  a  body  free  from  external  influences  would  come  to 
rest.  We  cannot,  of  course,  free  any  body  entirely  from  external 
influences;  but  we  can  greatly  diminish  these  influences,  and  with 


DYNAMICS  27 

each  diminution  the  velocity  becomes  more  nearly  constant.  The 
most  common  hindrance  to  steady  motion  is  friction.  A  stone 
given  a  push  along  a  rough  road  is  quickly  stopped  by  friction; 
on  a  smooth  floor  it  will  continue  longer  in  motion;  a  well- 
polished  stone  started  on  smooth  ice  will  continue  in  motion  for 
a  great  distance.  Such  considerations  make  it  seem  probable 
that,  if  freed  from  external  influences,  a  body  would  move  with 
constant  velocity;  they  do  not,  however,  amount  to  a  proof  of  the 
statement  in  the  first  law  of  motion.  The  proof  of  the  law  is  that 
all  of  the  innumerable  deductions  made  from  it  and  the  other 
laws  of  motion  are  verified  by  experience. 

The  law  implies  a  definition  of  force.  This  is  usually  given 
in  the  form  "Force  is  whatever  changes  or  tends  to  change  the 
motion  of  a  body"  or  "Force  is  that  which  produces  acceleration." 
Thus,  friction,  the  pull  of  a  stretched  spring,  the  attraction  of  the 
earth  on  a  body,  etc.,  are  forces;  when  a  body  revolves  in  a  circle, 
it  has  an  acceleration  toward  the  center  and  must,  therefore,  be 
acted  on  by  some  force.  What  exerts  a  force  on  a  body  is,  of 
course,  some  other  body.  Thus  the  friction  opposing  the  motion 
of  a  vehicle  is  due  to  the  earth  and  the  pull  of  a  spring  is  due 
to  the  spring,  etc.  The  word  "force"  is  therefore  a  name  which 
we  give  to  that  influence  of  one  body  on  another  by  which  the 
first  changes  the  motion  of  the  second. 

The  property  a  body  has  of  tending  to  persist  in  its  state  of 
motion  or  of  rest  is  called  Inertia. 

38.  The  Mass  of  a  Body. — Common  experience  shows  that,  when 
a  given  force  is  applied  to  a  body,  the  magnitude  of  the  accelera- 
tion depends  on  some  property  of  the  body.  Thus,  a  horizontal 
spring  kept  stretched  to  a  definite  length,  say  one  foot,  will  apply 
a  definite  force  to  the  body  to  which  it  is  attached.  If  attached 
to  a  cubic  foot  of  lead  supported  without  friction,  it  will  produce 
a  certain  acceleration;  but  the  acceleration  will  be  different,  if  a 
cubic  foot  of  wood  be  substituted  for  the  lead.  The  difference 
is  not  due  to  the  difference  in  the  weights  of  the  bodies,  since 
weight  is  a  force  that  acts  vertically  and  does  not  affect  the 
horizontal  motion  of  the  bodies.  The  difference  is  due  to  what 
we  call  the  masses  of  the  bodies. 

To  attach  a  definite  meaning  to  the  word  mass  we  must  define 
what  is  meant  by  the  ratio  of  the  masses  of  two  bodies.  The  ratio 


28          MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

of  the  masses  of  two  bodies  is  the  inverse  of  the  ratio  of  the  accelera- 
tions that  a  given  force  imparts  to  the  bodies  when  applied  to  them 
in  succession.  For  example,  if  a  body  A  acted  on  by  a  certain 
force  receives  twice  the  acceleration  that  a  second  body  B 
receives  when  acted  on  by  the  same  force,  the  mass  of  A  is  half 
as  great  as  the  mass  of  B  and  so  for  other  ratios.  Hence,  if  we 
adopt,  as  we  presently  shall,  a  certain  body  as  a  body  of  unit  mass, 
the  mass  of  any  other  body  becomes  definite. 

In  the  above  we  have  defined  the  ratio  of  two  masses  by  means  of  the 
ratio  of  the  accelerations  imparted  to  them  by  some  particular  force.  This 
at  once  suggests  the  question:  Would  the  ratio  be  found  different  if  some 
other  force  were  used  in  the  test?  If  so,  the  word  mass  as  applied  to  a  body 
would  have  no  definite  meaning  except  in  relation  to  a  particular  force. 
But,  as  a  matter  of  fact,  the  ratio  is  found  to  be  the  same,  no  matter  what  may 
be  the  force  chosen  for  the  test.  This  is  a  very  important  statement,  but  we 
do  not  need  to  state  it  as  a  separate  fundamental  principle  since  it  is  included 
in  Newton's  Second  Law  of  Motion  as  will  readily  be  seen  later  from  the 
formula  for  that  law  (§42).  The  fact  that  bodies  have  definite  masses,  the 
same  no  matter  what  their  accelerations  or  the  forces  acting  on  them,  was 
one  of  Newton's  most  important  discoveries. 

We  shall  see  later  (§570)  that  there  is  good  reason  to  believe  that  at 
immensely  great  velocities  the  mass  of  a  body  may  depend  appreciably  on 
its  velocity. 

Some  persons  find  difficulty  in  accepting  the  above  definition  of 
the  ratio  of  two  masses,  because  they  cannot  see  an  easy  means  of 
applying  it  directly  to  comparing  masses.  It  is,  however,  not 
given  as  a  practical  method  of  comparing  masses ;  but  it  leads,  as 
we  shall  see  later,  to  a  very  practical  method  (§42). 

39.  Units  of  Mass. — The  unit  of  mass  chiefly  employed  in 
Physics  is  the  gram,  which  is  defined  as  one  one-thousandth  of 
the  mass  of  a  block  of  platinum  kept  at  Sevres,  near  Paris,  and 
known  as  the  Kilogram  prototype.  Fractions  and  multiples  of  the 
gram  in  frequent  use  are  named  as  follows: 

Milligram  =.001  g.  Kilogram       =1000g. 

Centigram  =  .01     g.  f  =1,000,000  g. 

Decigram  =.1      g.  Metnc  ton  |  =  1000  kg. 

In  English-speaking  countries  the  pound  is,  for  commercial 
and  industrial  purposes,  used  as  unit  of  mass.  It  is  defined  as 
the  mass  of  a  certain  block  of  platinum  kept  at  the  Exchequer  in 


DYNAMICS  29 

London.     It  is  worth  remembering  that  1  kgm.  =2.20  Ibs.   ap- 
proximately and  that  1  pound  =  454  gms.  approximately. 

40.  Ratio  of  Forces. — Different  forces  applied  to  a  body  give  it 
different  accelerations.  For  example,  if  a  heavy  body  be  hung  from 
the  ceiling  by  a  cord  and  a  horizontal  cord  be  attached  to  it,  a 
small  pull  will  start  it  slowly,  while  a  stronger  pull  will  start  it 
more  rapidly.     Or,  if  a  horizontal  spiral  spring,  kept  stretched 
to  a  definite  length,  were  applied  to  a  body  supported  with  very 
little  friction  on  a  horizontal  table,  a  definite  acceleration  would 
be  produced.     If  this  experiment  were  repeated  with  the  spring 
stretched  to  a  different  length,  a  different  acceleration  would  re- 
sult.    These  illustrations  would  be  somewhat  difficult  to  carry 
out  accurately,  but  they  will  help  to  make  clear  the  following 
definition  of  the  ratio  of  two  forces,  and  from  this  we  shall  be 
able  to  deduce  a  more  accurate  method  of  finding  the  ratio, 
either  by  calculation  (§42)  or  by  static  experiments  (§52). 

The  ratio  of  two  forces  is  the  ratio  of  the  accelerations  they  can 
impart  to  a  given  body.  For  definiteness,  we  shall  suppose  that  the 
body  referred  to  is  one  of  unit  mass.  If  now  we  take  any  force 
as  unit  force,  the  magnitude  of  any  other  force  becomes  definite. 
For  simplicity,  we  shall  usually  take  as  unit  force  that  force  which, 
acting  on  unit  mass,  gives  it  unit  acceleration.  A  force  which 
gives  unit  mass  two  units  of  acceleration  will  then  be  a  force  of 
two  units,  and  so  on. 

In  the  above  we  have  defined  the  ratio  of  two  forces  by  the  ratio  of  the 
accelerations  they  impart  to  some  particular  body.  This  definition  would 
not  be  of  much  value,  if  the  ratio  obtained  depended  on  the  particular  body 
chosen.  As  a  matter  of  fact,  the  ratio  of  two  forces,  as  defined  above,  is  the 
same,  no  matter  what  body  is  chosen  for  the  test.  This  statement,  while 
very  important,  does  not  need  to  be  stated  as  a  separate  fundamental  prin- 
ciple, since  it  is  included  in  Newton's  Second  Law  of  Motion,  as  will  readily 
be  seen  later  from  the  formula  for  that  law  (§42). 

41.  Momentum. — Every  one  is  aware  that  certain  properties 
of  moving  bodies  depend  on  mass  and  velocity  conjointly.     Thus, 
the  length  of  time  required  by  a  locomotive  to  start  a  train 
depends  on  both  the  mass  of  the  train  and  the  velocity  to  be 
imparted  to  it,  and  the  same  is  true  of  stopping  it.     Hence  we 
find  it  convenient  to  define  a  property  depending  on  mass  and 
velocity  conjointly.     Momentum  is  defined  as  the  product  of  mass 


30          MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

and  velocity.  Since  the  velocity  has  direction  as  well  as  magni- 
tude, while  the  mass  has  magnitude  only,  the  momentum  of  a  body 
is  a  vector  quantity,  the  direction  of  which  is  that  of  the  velocity. 
(What  we  now  call  momentum  Newton  called  quantity  of  motion 
as  distinguished  from  rate  of  motion  or  velocity.) 

When  the  velocity  of  a  body  changes,  its  momentum  also 
changes.  Since  the  mass  of  the  body  is  constant,  any  change  in 
the  momentum  of  a  body  must  be  due  to  a  change  of  its  velocity, 
and  the  change  of  momentum  must  equal  the  product  of  the 
mass  and  the  change  of  velocity.  Hence,  when  the  momentum 
of  a  body  is  changing,  the  rate  of  change  of  momentum  equals 
the  product  of  the  mass  and  the  rate  of  change  of  velocity,  that 
is,  the  product,  ma,  of  the  mass  m  and  its  acceleration  a. 

42.  Newton's  Second  Law  of  Motion.  —  The  rate  of  change  of 
the  momentum  of  a  body  is  proportional  to  the  force  acting  on  the 
body  and  is  in  the  direction  of  the  force. 

To  reduce  this  statement  to  a  mathematical  formula,  let  us 
suppose  that  a  force  Fl  acting  on  a  mass  ml  gives  it  an  accelera- 
tion alt  and  that  a  force  F2  acting  on  a  mass  w2  gives  it  an  accel- 
eration a2,  and  so  on  for  any  number  of  forces  and  masses. 
Then 


If  k  be  used  to  denote  the  constant  ratio  of  F  to  ma 

F  =  kma 

This  is  the  general  formula  for  Newton's  Second  Law.  The 
constant  k  is  a  number  the  magnitude  of  which  depends  on  the 
unit  chosen  for  Ft  since  we  have  already  chosen  certain  units 
for  m  and  a. 

Since,  as  Galileo  found  (and  as  Newton  and  Bessel  proved 
more  completely),  all  bodies  fall  with  the  same  acceleration 
(allowance  being  made  for  air  friction),  it  follows  from  the 
above  formula  that  the  masses  of  bodies  are  proportional  to  their 
weights.  This  is  the  principle  of  the  common  balance,  by  which 
the  masses  of  bodies  are  compared  by  comparing  their  weights. 

43.  Units  of  Force.  —  (1)  Absolute  Units.  —  Calculations  by 
means  of  the  formula  for  Newton's  Second  Law  of  Motion  are 
much  simplified  when  the  unit  of  force  is  so  chosen  that  k  is  1. 
Such  a  unit  of  force  is  called  an  absolute  unit  of  force. 


DYNAMICS  31 

Since  F  is  to  be  1  when  m  and  a  are  each  1,  an  absolute  unit  of 
force  is  that  force  which,  acting  on  a  body  of  unit  mass  gives  it 
unit  acceleration.  Hence 

F  =  ma  in  absolute  units. 

If  the  gram  be  taken  as  unit  of  mass  and  the  cm.  per  sec.  per 
sec.  as  unit  of  acceleration,  the  absolute  unit  of  force  is  that  force 
which,  acting  on  a  body  of  one  gram  mass,  gives  it  an  acceleration 
of  one  cm.  per  sec.  per  sec.  and  is  called  a  dyne. 

If  the  pound  be  taken  as  unit  of  mass  and  the  foot  per  sec. 
per  sec.  as  unit  of  acceleration,  the  absolute  unit  of  force  is  that 
force  which,  acting  on  a  body  of  one  pound  mass,  gives  it  an 
acceleration  of  one  foot  per  sec.  per  sec.  and  is  called  a  poundal. 

(2)  Gravitational  Units. — The  weight  of  a  body  of  unit  mass  is, 
for  many  purposes,  a  convenient  unit  of  force;  but,  when  it  is 
chosen,  the  value  of  k  is  not  1.  For  allow  a  body  of  unit  mass  to 
fall:  the  acceleration  a  =  g,  while  F  =  1  and  m  =  1.  Hence, 
substituting  in  F  =  kma,  we  get  1  =  kg,  and  therefore  k  =  1/g. 
Hence 

F  =  —a  in  gravitational  units. 

The  only  gravitational  unit  we  need  consider  is  the  weight  of 
a  pound.  With  the  foot  per  sec.  per  sec.  as  unit  of  acceleration, 
the  value  of  g  is  32.2  approximately  but  varies  with  the  locality. 
Formulae  derived  from  the  above  will  always  have  m/g  where  m 
only  would  appear  in  absolute  units. 

Engineers  prefer  to  write  W,  the  number  of  pounds  weight  in  the  weight 
of  a  body,  instead  of  m,  the  number  of  pounds  mass  in  the  mass  of  a  body. 
The  two  are  equal  numerically. 

44.  Newton's  Second  Law  (Continued). — The  statement  of  the 
second  law  of  motion  is  so  brief  that  some  things  implied  in  it 
might  easily  escape  notice: 

1.  In  the  statement  of  the  law  the  rate  of  change  of  momentum 
of  a  body  is  spoken  of,  without  any  reference  to  whether  the  body 
starts  from  rest  or  is  initially  in  motion.  Hence  it  is  implied 
that  the  effect  of  a  force  applied  to  a  body  is  independent  of  the  state 
of  motion  of  the  body  when  the  force  begins  to  act.  For  example, 
gravity  is  a  force  that  acts  vertically  downward.  When  a  body 
is  dropped  from  a  height,  the  force  of  gravity  gives  it  a  certain 


32          MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

acceleration  downward;  if  the  same  body  be  started  downward 
with  a  certain  velocity,  its  acceleration  downward  will  be  the 
same  as  when  the  body  is  simply  dropped,  and  the  same  will  be 
true  if  the  body  be  given  an  initial  velocity  upward  or  in  any 
direction.  It  is  found  possible  to  play  games  of  ball  or  cricket 
on  a  moving  steamship;  the  effect  of  throwing  the  ball  with  a 
certain  force  or  striking  it  with  a  bat  is  the  same  as  when  the 
steamship  is  at  rest. 

2.  The  law  states  how  a  force  will  affect  the  motion  of  a  body, 
but  it  makes  no  reference  to  whether  some  other  force  is  acting 
on  the  body  at  the  same  time  or  not.  Hence  it  is  implied  that 
each  force  produces  its  own  effect  independently  of  the  simultaneous 
action  of  any  other  force;  and,  when  several  forces  act  on  a  body, 
we  may  calculate  the  acceleration  produced  by  each  as  if  the 
other  forces  did  not  exist,  and  then  add  the  accelerations  to 
find  the  whole  effect  of  all  the  forces.  This  very  important  prin- 
ciple is  sometimes  called  that  of  the  independence  of  forces. 

45.  Impulse  of  a  Force.  —  The  product  of  a  force  and  the  time 
during  which  it  acts  is  called  the  impulse  of  the  force.  When  a 
force  F  acts  on  a  mass  m  for  time  t,  from  the  formula  for  the 
Second  Law  of  Motion,  by  multiplying  both  sides  by  t,  we  get: 

Ft  —  mat 

Now  at  is  the  increase  of  velocity  produced,  and  this,  multiplied 
by  m,  is  the  increase  of  momentum.  Hence  the  impulse  of  a  force 
equals  the  momentum  produced  by  it.  If  the  body,  starting  with 
a  velocity  v0,  has  at  time  t  a  velocity  v, 


46.  Newton's  Third  Law  of  Motion.  —  Action  and  reaction  are 
equal  and  opposite.  In  the  statements  of  the  first  and  second 
laws  of  motion  forces  acting  on  bodies  are  spoken  of,  but  nothing 
is  said  as  to  what  exerts  force.  This  lack  is  supplied  by  the 
third  law. 

The  action  and  reaction  here  referred  to  mean/orce  and  counter- 
force.  The  meaning  of  the  statement  is  that  force  on  any  one 
body  is  exerted  by  some  other  body,  and  this  other  body  itself 
experiences  an  equal  and  opposite  force  exerted  by  the  first 
body,  the  line  of  action  of  both  forces  being  the  line  joining  the 
two  bodies. 


DYNAMICS  33 

In  many  cases  the  truth  of  this  law  will  be  recognized  as  being 
evident.  For  example,  when  one  presses  his  two  hands  against 
each  other,  it  will  be  admitted  that  the  hands,  if  at  rest,  press 
equally  in  opposite  directions.  If  one  hand  be  pressed  against  a 
wall,  the  same  must  still  hold,  since  the  wall  merely  takes  the 
place  of  the  other  hand  in  the  first  illustration.  But  the  case  is 
not  so  clear  when  a  hand  is  pressed  against  an  obstacle  that 
moves.  How,  it  is  sometimes  asked,  can  there  be  motion 
produced  if  the  forces  are  equal  and  opposite?  The  answer  is 
that  the  two  forces  spoken  of  do  not  act  on  one  body;  there  is  one 
force  exerted  by  the  hand  on  the  obstacle,  and  the  obstacle  yields 
unless  restrained  by  some  other  force;  the  reaction  is  the  back 
pressure  of  the  body  on  the  hand,  not  a  force  acting  on  the  body. 

Consider,  also,  the  forces  that  come  into  play  when  a  horse  of  mass  ml 
pulling  on  a  horizontal  rope  of  mass  ma  draws  a  block  of  mass  m,.  Here  there 
are  four  pairs  of  actions  and  reactions.  In  the  first  place,  the  horse  pushes 
against  the  ground  and  the  reaction  of 
the  ground  is  an  equal  and  opposite 
push.  Let  the  magnitude  of  this  hori- 
zontal action  and  reaction  be  Ft. 
Secondly  the  horse  exerts  a  forward 


pull,  of  magnitude  say  Fv  on  the  Fia'  16'~ 
rope  and  the  reaction  of  the  rope  is 
equal  and  opposite.  The  rope  exerts  a  horizontal  force  on  the  block  and 
the  block  exerts  an  equal  and  opposite  reaction,  the  magnitude  of  each 
being  Ft.  Finally,  there  is  the  action  and  reaction  between  the  block  and 
the  ground;  let  the  horizontal  component  of  this  have  a  magnitude  Ft. 
If  there  is  an  acceleration  a,  as  there  must  be  to  begin  the  motion,  Fl  is 
greater  than  Ft  by  m^,  Ft  is  greater  than  Fs  by  maa,  and  Ft  is  greater  than 
Ft  by  m,a.  Thus  Fl  exceeds  F4  by  (m,-|-7n2-f-m8)a,  and  this  is,  therefore, 
the  total  backward  push  on  the  ground.  When  the  motion  has  become 
constant,  a  =  o  and  all  the  forces  mentioned  are  of  equal  magnitude. 

Since  a  force  is  always  accompanied  by  a  counterforce,  the  two 
are  parts  or  different  aspects  of  one  inseparable  whole,  and  the 
two  together  constitute  what  is  called  a  stress.  Thus  every 
force  is  the  partial  aspect  of  some  stress,  just  as  a  purchase  and  a 
sale  are  partial  aspects  of  an  exchange. 

47.  Force  Required  for  Motion  in  a  Circle. — When  a  particle 
revolves  in  a  circle,  it  has  an  acceleration  toward  the  center 
equal  to  v*/r  (§  32),  where  v  is  the  magnitude  of  the  velocity  (i.e. , 
the  speed)  and  r  is  the  radius.  To  cause  this^acceleration  there 


34          MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

must  be  a  force  directed  toward  the  center,  and,  according  to 
Newton's  Second  Law,  this  centripetal  force  F  must  be  such  that 


Against  this  force  the  particle  will  exert  an  equal  and  opposite 
reaction  on  the  body  that  exerts  the  force  toward  the  center. 
If,  for  example,  the  particle  be  attached  by  a 
string  to  the  finger,  the  reaction  will  be  a  force 
acting  on  the  finger  and  will  be  in  a  direction 
outward  along  the  radius.      This  reaction  is 
called  a  centrifugal  force.     Thus  the  centrifugal 
force  is  not  a  force  acting  on  the  moving  particle, 
but  a  reaction,  exerted  by  the  particle,  on  the 
FIG.  IT.-A  particle     other  body  that  exerts  tfo  jorce  toward  fa  center. 

moving  m  a  circle  is  J  * 

acted  on  by  a  force     (That  the  above  formula  also  applies  to  the 
toward  the  center.          motion  of  a  body  is  shown  in  §101.) 

Illustrations  of  the  above  are  very  numerous  and  a  few  may  be  mentioned. 
Drops  of  water  are  thrown  off  tangentially  from  a  rapidly  moving  bicycle  or 
carriage  wheel,  owing  to  the  fact  that  there  is  not  a  sufficient  force  toward 
the  center  acting  on  them,  and  they  therefore  move  off  on  a  tangent,  in 
accordance  with  the  first  law  of  motion.  A  train  rounding  a  curve  presses 
outward  on  the  rails,  and  the  resultant  of  this  force  and  the  vertical  weight 
of  the  train  is  a  force  inclined  to  the  vertical.  Since  it  is  desirable  that  the 
whole  force  should  be  perpendicular  to  the  sleepers,  the  outer  rail  is  raised. 
In  the  Centrifugal  drier,  used  in  laundries  and  sugar  refineries,  the  material 
to  be  dried  is  placed  in  a  perforated  cylinder  rotating  about  its  axis  which  is 
vertical;  the  drops  of  water,  not  being  held  by  a  force  directed  to  the  center, 
escape  through  the  perforations.  In  the  Centrifugal  cream-separator,  which 
is  a  rotating  vertical  cylinder,  both  the  milk  and  the  cream  tend  to  move  as 
far  from  the  axis  as  possible;  but  the  milk,  being  the  denser,  exerts  the  more 
powerful  tendency  and  therefore  occupies  the  parts  of  the  vessel  farthest 
from  the  axis.  The  flattening  of  the  earth  at  its  poles  is  due  to  its  axial 
rotation;  if  at  rest  it  would  be  spherical;  but,  being  in  rotation,  it  bulges  at 
the  equator  to  such  an  extent  that  the  restoring  forces  due  to  gravitational 
attractions  supply  the  requisite  force  toward  the  center.  The  higher  the 
speed  of  belting  the  less  it  presses  on  a  pulley  and  the  more  liable,  there- 
fore, it  is  to  slip;  for  more  of  the  tension  of  the  belting  is  called  on  to  supply 
the  requisite  force  toward  the  center.  Watt's  governor  for  a  steam-engine 
consists  of  a  pair  of  balls  whirled  around  a  vertical  spindle  at  a  rate  pro- 
portional to  the  speed  of  the  engine;  when  this  speed  exceeds  the  desired 
limit  the  outward  movement  of  the  balls  acts  on  a  steam-valve  so  as  to 
decrease  the  speed  of  the  engine. 


DYNAMICS  35 

Resultant  of  Forces. — Equilibrium 

48.  Composition  of  Forces. — Two  or  more  forces  may  act  on  a 
body  at  the  same  time.  For  example,  a  body  falling  because  of 
the  attraction  of  the  earth  may  be  drawn  horizontally  by  a 
stretched  spring  or  blown  by  wind  pressure.  In  such  cases  each 
force  produces  an  acceleration  independently  of  the  action  of  the 
other  forces  (§44),  and  the  body  travels  in  some  path  with  a 
definite  acceleration,  which  is  the  resultant  of  the  accelerations 
produced  by  the  separate  forces. 

The  resultant  of  two  or  more  forces  is  defined  as  the  single  force 
which  will  produce  the  resultant  acceleration.  The  resultant  of 
any  number  of  forces  which  act  on  a  particle  can 
be  found  by  vector  addition,  that  is  by  a  tri- 
angle, parallelogram,  or  polygon  construction. 
For  a  force  has  a  certain  magnitude  and  a  certain 
direction  and  is,  therefore,  a  vector  quantity. 
Hence  any  number  of  forces  acting  on  a  particle 
may  be  represented  by  lines  drawn  from  a  point.  Let  AB  and 
AC  represent  two  forces  Fl  and  F2,  acting  on  a  particle.  Com- 
plete the  parallelogram  ABDC.  By  the  Second  Law  of  Motion 
the  accelerations  produced  by  F,  and  F2  are  in  the  directions  of 
and  proportional  to  AB  and  AC,  and  the  resultant  acceleration 
must,  therefore,  be  represented  by  AD;  and,  since  the  resultant 
force  is  the  force  that  will  produce  the  resultant  acceleration,  it 
must  be  in  the  direction  AD*  If,  now,  we  denote  the  accelera- 
tions produced  by  Fl  and  F2  by  al  and  a2  respectively  and  if 
the  resultant  force  and  acceleration  be  denoted  by  F  and  a 
respectively,  by  the  Second  Law  of  Motion 
F  :  Ft  :  F2  :  :  ma  :  mal  :  ma2 

:  :  a  :al  :  a2 

::AD:AB: AC 

Hence  AD  represents  the  resultant  force  on  the  scale  on  which 
AB  and  AC  represent  the  separate  forces.  This  very  important 
result,  called  the  Parallelogram  of  Forces,  is  usually  stated  as 
follows: 

//  two  forces  acting  on  a  particle  be  represented  by  two  lines 
drawn  from  a  point  and  if  a  parallelogram  be  drawn  with  these  two 


36 


MECHANICS  AND  THE  PROPERTIES  OF  MATTER 


lines  as  sides,  the  resultant  will  be  represented  by  the  diagonal  that 
passes  through  the  point. 

Since,  then,  we  may  add  two  forces  by  the  parallelogram  method 
or  by  the  triangle  method  (which  is  essentially  the  same),  we 
may  in  the  same  way  add  a  third  to  the  resultant  of  these  two 
and  so  on.  Hence  the  polygon  method  of  addition  applies  to 
forces  acting  on  a  particle. 

Let  6  be  the  angle  between  the  directions  of  the  forces  Fl  and 
7^,.  Then,  as  in  the  case  of  velocities  and  accelerations, 

2*  +  2FlF2cos  6 


49.  Resolution  of  a  Force  into  Components.  Since  two  or 
more  forces  acting  on  a  particle  can  be  replaced  by  a  single  force 
called  their  resultant,  a  single  force  can  be  replaced  by  any  two 
or  more  forces  which,  added  geometrically,  give  the  single  force. 
This  is  called  the  resolution  of  a  force  into  components. 

The  most  important  case  practically  is  when  the  components 
are  at  right  angles  to  each  other.  When  a  single  force  is  resolved 
into  two  components  (Fig.  19),  the  components  and  the  force  re- 
solved must  be  in  the  same  plane.  When  the  two  components  are 

Y 
B 


^z 


F  cos  ct 
Fio.  19. 


Fio.  20. 


at  right  angles,  the  component  that  makes  an  angle  a  with  the  whole 
original  force  F  has  a  magnitude  F  cos  a,  and  the  other  component 
is  F  sin  a.  The  agreement  as  regards  the  signs  of  angles  noted 
in  §25  applies  to  the  present  case. 

A  force  F  may  also  be  resolved  into  three  components  in  three 
directions  at  right  angles  to  each  other.  All  that  is  necessary 
is  to  construct  a  right-angled  parallelepiped  with  the  line  represent- 
ing F  as  diagonal  (Fig.  20)  and  with  edges  in  the  three  rectangular 
directions.  If  the  three  directions  be  taken  as  axes  of  x,  y  and 
z  and  if  the  components  be  denoted  by  Fx,  Fv,  Fz  respectively, 
we  shall  have 


DYNAMICS 


37 


60.  Illustrations  of  the  Resolution  of  a  Force  into  Components. — 1.  The 

force  of  gravity  on  a  body  of  mass  m  acts  vertically  downward  and,  in  absolute 
units,  equals  mg.  If  a  body  is  not  free  to  move  vertically,  but  is  free  to 
move  in  some  other  direction,  the  only  part  of  gravity  that  can  affect  the 
motion  is  the  component  in  that  direction.  For 
instance,  11  a  body  (Fig.  21)  be  on  a  smooth  plane 
inclined  at  an  angle  i  to  the  horizontal,  the  force  of 
gravity,  mg,  may  be  resolved  into  a  component 
mg  sin  i  down  the  plane  and  a  component  mg  cos  i 
perpendicular  to  the  plane.  The  latter  component 
will  produce  pressure  on  the  plane  but  will  not 
affect  the  motion  down  the  plane,  which  will  de- 
pend only  on  the  former  component,  mg  sin  t.  If 
the  plane  be  not  perfectly  smooth,  there  will  also 

be  a  force  of  friction,  say  F,  parallel  to  the  plane,  and  the  resultant  force 
down  the  plane  will  be  (mg  sin  i—F). 

2.  A  sail-boat  (Fig.  22)  effects  a  double  resolution  of  the  wind  pressure.  The 
component  of  the  wind  pressure  W  parallel  to  the  plane  of  the  sails  has  very 
little  effect ;  the  component,  say  F,  perpendicular  to  the  sail  is  the  effective  com- 
ponent. Again,  F  may  be  resolved  into  a  component  perpendicular  to  the 
keel  and  a  component  /  parallel  to  the  keel.  The  former  produces  a  small 
sidewise  motion  or  lee-way,  while  the  latter,  being  in  the  direction  in  which 
the  boat  is  most  free  to  move,  is  the  effective  component. 


Fio.  22. 


Fio.  23. 


3.  In  the  case  of  a  kite  (Fig.  23)  the  component  of  the  wind  pressure  parallel 
to  the  surface  of  the  kite  has  no  effect;  the  component  perpendicular  to  the 
kite,  when  the  kite  has  risen  to  the  proper  level,  is  equal  and  opposite  to 
the  resultant  of  the  pull,  T,  of  the  cord  and  the  weight,  mg,  of  the  kite. 

61.  Analytical  Method  of  Compounding  Forces. — A  simple  and 
general  method  of  finding  the  resultant  of  a  number  of  forces  in  a 
plane  is  to  resolve  each  in  two  directions  at  right  angles  and  then 
add  all  these  components.  Thus,  let  Flt  F,  •  •  •  be  the  forces 
acting  on  a  particle  at  0.  Take  any  two  convenient  rectangular 
directions,  Ox  and  Or/,  and  let  the  angles  Flt  Fa  •  •  •  make  with 
Ox  be  alf  «3  •  •  •  respectively.  Then  Fl  is  equivalent  to  Fl  cos  at 


38 


MECHANICS  AND  THE  PROPERTIES  OF  MATTER 


along  Ox  and  Fl  sin  «1  along  Oy  and  so  for  the  other  forces. 
Let  the  sum  of  the  components  along  Ox  be  denoted  by  X  and 
the  sum  of  the  components  along  Oy  by  Y.  Then 


Fj  cos  al-\-F2  cos  a2+  -  - 
Fl  sin  at  +  F2  sin  a2  +  •  • 


=  I  F  cos  a 
=  I  F  sin  a 

We  have  thus  replaced  the  forces  FltF2-  •  -by  X  along  Ox  and  Y 
along  Ot/.  The  resultant  of  X  and  Y  is  the  resultant  of  F19  F2, 
etc.  Let  the  magnitude  of  the  resultant  be  R  and  let  it  make 
an  angle  6  with  OX.  Then 


tan  6 


Y 

X 


These  formulae  give  the  magnitude  and  the  direction  of  the  result- 
ant. In  using  .this  method  it  must  be  remembered  that,  when 
we  substitute  for  each  angle  a  its  numerical  value,  we  must  call 
the  angle  positive  if  it  is  measured  in  the 
direction  regarded  as  positive,  say  the 
counter-clockwise  direction;  if  measured  in 
the  opposite  direction  it  must  be  regarded 
as  negative. 

The  angle  6  that  the  resultant  makes 
with  Ox  is  found  from  its  tangent.  When 
the  tangent  is  positive,  it  shows  that  the 
angle  is  between  0  and  90°  or  between  180° 
and  270°.  To  decide  between  these  two, 
note  that  the  signs  of  the  values  of  X  and 
Y  must  be  either  both  positive  or  both 
negative,  since  the  tangent  is  positive.  If 
both  are  positive,  0  is  between  0°  and  90°; 
if  both  are  negative,  it  lies  between  180° 
and  270°.  The  reader  should  have  no  diffi- 
culty in  completing  the  reasoning  for  the  case  in  which  tan  0  is 
negative. 

When  X  and  Y  are  both  zero,  that  is,  when  the  sum  of  the  com- 
ponents in  each  of  two  directions  at  right  angles  is  zero,  R  is 
also  zero.  Conversely,  when  R  is  zero,  X  and  Y  must  also  each 
be  zero,  since  the  square  of  a  number  cannot  be  negative. 


0      X 

FIG.  24. — Analytical 
method  of  compounding 
forces. 


DYNAMICS  39 

When  the  forces  to  be  compounded  are  not  all  in  one  plane, 
we  may  take  three  directions,  Ox,  Oy,  Oz,  at  right  angles  and 
resolve  each  force  into  components  in  these  three  directions. 
Denote  the  sum  of  the  components  along  Ox  by  X,  that  along 
Oy  by  7,  and  that  along  Oz  by  Z  and  let  the  resultant  be  R. 
Then 


If  X  =  Q,  F  =  0  and  Z  =  0,  then  #  =  0.  The  converse  is  also 
true,  since  X2,  Y2  and  Z2  must  be  either  positive  or  zero. 

62.  Equilibrium  of  Forces  Acting  on  a  Particle.  —  When  the 
resultant  of  the  forces  acting  on  a  particle  is  zero, 
the  forces  are  said  to  be  in  equilibrium,  that  is,  in 
a  state  of  balance,  so  that  they  do  not  change 
the  motion  of  the  particle. 

When  two  equal  and  opposite  forces  act  on  a 
particle,  they  are  in  equilibrium,  for  their  re- 
sultant is  zero.  Conversely,  if  two  forces  are  in 
equilibrium,  they  must  be  equal  and  opposite, 
for  otherwise  their  resultant  could  not  be  zero. 

When  three  forces  acting  on  a  particle  are  in  the  direction  of 
and  proportional  to  the  sides  of  a  triangle  taken  in  order,  they  are  in 
equilibrium.  For  if  the  three  forces  Flt  F2,  F3  be  in  the  direc- 
tion of  and  proportional  to  A  B,  BC,  CA,  the  resultant  of  Fl  and 
F2  will  be  represented  by  AC,  and  the  resultant  of  forces  repre- 
sented by  1C  and  CA  is  zero.  The  converse  of  this  proposition 


^ 

A'< 


'• 

FIG.  26. 

is  that,  if  three  forces  acting  on  a  particle  be  in  equilibrium,  and 
if  any  triangle  be  drawn  with  its  sides  respectively  in  the  directions  of 
the  forces,  the  forces  will  be  proportional  to  the  sides  of  this  tri- 
angle. To  prove  this  let  us  suppose  that  A  B  and  BC  (Fig.  26), 
are  any  two  lines  in  the  direction  of  and  proportional  to  two  of 


40          MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

the  forces  Fl  and  Fy  Then  a  force  represented  by  AC  is  equiva- 
lent to  jPt  and  F2  taken  together.  Hence,  since  the  forces  are  in 
equilibrium,  the  third  force  Fs  must  be  in  the  direction  of  and 
proportional  to  CA.  Now  any  other  triangle  such  as  A'5'C", 
with  sides  in  the  directions  of  Flt  F2,  F3  respectively,  that  is,  in 
the  directions  of  A B,  BC,  CA,  respectively,  must  be  similar  to 
ABC.  Hence  its  sides  must  be  proportional  to  the  sides  of  ABC, 
that  is,  to  Fl9  F2,  F3  respectively.  This  converse  proposition  is 
very  important,  for,  when  we  know  the  directions  of  three  forces 
that  are  in  equilibrium,  we  can  find  the  relative  magnitudes  of 
the  forces  by  constructing  a  triangle  with  its  sides  in  the  direc- 
tions of  the  forces. 

When  any  number  of  forces  acting  on  a  particle  are  in  the 
directions  of  and  proportional  to  the  sides  of  a  closed  polygon 
taken  in  order,  they  are  in  equilibrium ;  for  the  resultant  is  zero. 
The  converse  of  this  proposition,  for  more  than  three  forces,  is 
not  true;  for  polygons  are  not  necessarily  similar  when  their 
respective  sides  are  parallel. 

When  any  number  of  forces  are  such  that  the  sum  of  their  com- 
ponents in  each  of  three  directions  at  right  angles  is  zero,  they 
are  in  equilibrium.  This  is  evident  from  §51,  for,  when  X,  Y  and 
Z  are  all  zero,  R  must  also  be  zero.  Conversely,  when  any  num- 
ber of  forces  are  in  equilibrium,  the  sum  of  their  components  in 
any  direction  equals  zero ;  for  we  may  take  this  direction  as  one  of 
three  at  right  angles;  and,  since  R  is  zero,  the  sum  of  the  com- 
ponents in  each  of  these  directions  is  zero  (§51). 

Work  and  Energy 

53.  Work. — When  a  force  acts  on  a  body,  the  product  of  the 
force  by  the  distance  through  which  it  acts  in  the  direction  of  the 
force  is,  as  we  shall  see  later,  a  very  important  quantity  and  is 
called  the  work  performed  by  the  force.  Thus,  when  a  force  ap-  • 
plied  to  a  heavy  body  raises  it  a  certain  vertical  distance,  work  is 
performed  by  the  force,  the  amount  of  the  work  being  the  product 
of  the  force  and  the  distance  of  ascent;  and,  when  a  horizontal 
force  draws  a  body  horizontally,  the  work  is  the  product  of  the 
force  and  the  horizontal  distance. 

The  phrase  "in  the  direction  of  the  force"  that  occurs  in  the 


DYNAMICS  41 

definition  of  work  should  be  carefully  noted.  When  there  is  no 
motion  in  the  direction  of  a  force,  no  work  is  performed  by  that 
force.  For  instance,  a  travelling  crane  may  by  its  chains  exert 
an  enormous  force  in  sustaining  a  heavy  body  and  it  may  move 
the  body  through  a  great  distance  horizontally,  but  the  force 
exerted  by  the  chains  will  do  no  work  if  there  is  no  vertical 
motion.  If  a  force  F  acts  constantly  on  a  body  while  the  body 
moves  a  distance  AB  which  is  not  in  the  direction  of  the  force, 
to  get  the  work  performed  we  must  take  the  projection  of  A B  on 
the  line  of  action  of  the  force  and  multiply  the  projection  by  the 
force.  If  6  is  the  angle  between  AB  and  the  direction  of  the 
force,  the  proj  ection,  A  C,  of  A  B  on  the  line  of  the  action  of  the  force 
is  AB  cos  0  and  the  work  performed  is  F  •  AB  cos  6 .  This  at 
once  suggests  another  method  of  calculating  the  work  performed, 
for  F  •  AB  cos  0  is  the  same  as  F  cos  6-  AB  and  F  cos  6  is  the 


Fio.  27.  Fio.  28. 

component  of  F  in  the  direction  of  AB.  Thus  the  work  performed 
by  a  force  is  also  the  product  of  the  total  distance  by  the  com- 
ponent of  the  force  in  the  direction  of  motion. 

Work,  or  the  product  of  force  and  distance,  must  be  carefully 
distinguished  from  the  impulse  of  a  force,  which  is  the  product  of 
a  force  and  the  time  during  which  it  acts  (§45).  Given  a  force 
and  the  distance  through  which  it  acts,  we  do  not  need  to  know 
the  time  in  order  to  calculate  the  work. 

64.  Positive  and  Negative  Work. — Forces  always  exist  in  pairs 
of  equal  and  opposite  forces  (§46).  Hence,  when  a  force  applied 
to  a  body  does  work  by  moving  the  body  in  the  direction  of  the 
force,  it  must  at  the  same  time  overcome  an  opposing  force  or 
reaction.  The  applied  force  in  this  case  does  positive  work,  since 
the  motion  is  in  the  direction  of  the  applied  force.  This  work 
is  done  against  the  reaction,  or  we  may  say  that  the  reaction  does 
negative  work,  since  the  motion  is  in  the  opposite  direction  to  the 
reaction. 

The  nature  of  the  reaction  is  different  in  different  cases.     A 


42 


MECHANICS  AND  THE  PROPERTIES  OF  MATTER 


horse  attached  to  a  wagon  is  doing  work  against  the  force  of 
friction  when  the  wagon  is  moving  uniformly,  and  the  force  of 
friction  does  negative  work.  In  starting  the  wagon  into  motion 
the  horse  does  work  against  the  inertia  of  the  wagon  and  also 
of  the  horse,  in  addition  to  the  work  it  does  against  friction. 
When  a  body  is  moving  in  one  direction  and  a  force  is  suddenly 
applied  to  it  in  the  opposite  direction,  the  body  does  positive 
work  against  the  force,  which  in  this  case  does  negative  work. 

55.  Units  of  Work.-^-The  unit  of  work  is  the  work  done  by  the 
unit  force  in  acting  through  unit  distance.     When  the  dyne  is 
taken  as  unit  of  force  and  the  cm.  as  unit  of  length,  the  unit  of 
work  is  that  performed  by  a  dyne  acting  through  a  cm.  and  is 
called  an  erg.    Since  this  is  a  very  small  unit,  a  multiple  of  it, 
namely  10,000,000,  (or  107)  ergs,  is  frequently  used  and  is  called 
a  joule. 

When  the  weight  of  a  pound  is  taken  as  unit  of  force  and  the 
foot  as  unit  of  length,  the  unit  of  work  is  the  work  done  by  a 
force  equal  to  the  weight  of  one  pound  when  it  acts  through 
one  foot  and  is  called  a  foot-pound.  The  work  done  by  a 
poundal  acting  through  a  foot  is  called  a  foot  poundal. 

56.  Diagram  of  Work. — When  a  force  is  constant,  to  find  the 
work  it  does  we  multiply  the  magnitude  of  the  force  by  that  of 
the  displacement;  but,  when  a  force  is  variable,  some  other  method 
has  to  be  adopted.     One  way  is  to  divide  the  whole  displacement 
up  into  small  parts  and  multiply  each  small  part  by  the  force  at 

the  middle  of  the  small  displacement  and 
then  add  all  the  products.  Stated  briefly, 
JF  =  ^F.As.  By  taking  the  parts  small 
enough  we  may  get  the  work  as  accurately  as 
may  be  desired.  A  graphical  method  is  often 
preferable.  It  is  entirely  similar  to  the  method 
used  in  finding  the  distance  a  point  travels 
when  it  has  a  variable  velocity  (§22).  Let 
OA  be  a  line  that  represents,  on  some  scale,  the 
whole  displacement  measured  in  the  direction 
Divide  OA  into  a  very  large  number  of  very 
At  0  erect  a  perpendicular  OB  to  represent, 


0         a  b        A 

Fio.  29.— Diagram  of 
work. 


of  the  force. 

small  equal  parts. 

on  some  scale,  the  force  at  the  beginning  of  the  first  part;  erect 

similar  perpendiculars  to  represent  the  magnitude  of  the  force 


DYNAMICS  43 

at  the  beginning  of  the  other  parts,  and  through  the  ends  of  these 
perpendiculars  draw  a  smooth  curve  BC  If  we  calculated  the 
work  done  in  a  small  displacement  ab  by  taking  for  the  force  its 
value  at  the  beginning  of  ab,  the  result  would  be  too  small ;  and,  if 
we  made  the  calculation  by  taking  the  value  of  the  force  at  the 
end  of  ab,  the  result  would  be  too  large,  and  similarly  for  all  the 
other  intervals.  By  continuing  the  reasoning  as  in  §22,  we  find 
that  the  actual  work  done  is  represented  by  the  area  OBCA. 

Thus,  to  find  the  whole  work,  we  need  only  to  measure  the  area 
of  the  figure  and  then  allow  for  the  scale  on  which  it  is  drawn. 
If  each  unit  of  length  along  OA  stands  for  m  units  of  length  in 
the  displacement,  and  if  each  unit  of  length  along  OB  stands 
for  n  units  of  force,  each  unit  of  area  will  stand  for  mn  units  of 
work,  and  the  whole  area  multiplied  by  mn  will  give  the  whole 
work. 

When  the  curve  of  force  is  a  straight  line  the  area  may  be 
readily  calculated.  For  example,  let  us  calculate  the  work  done 
in  stretching  a  spring.  In  this  case  it  is  known  that  the  force 
that  is  needed  to  keep  a  spring  stretched  is  proportional  to  the 
amount  of  the  stretch  or  increase  of  length  (provided  this  be  not 
so  great  as  to  permanently  lengthen  the  spring).  Hence,  if  the 
spring  is  stretched  by  an  amount  x,  the  force  applied  to  it  is  kx, 
where  k  is  a  constant  and  is  evidently  equal  to  the  force  required 
to  produce  unit  increase  of  length.  If,  then,  a  curve  be  drawn 
with  the  values  of  kx  as  ordinates  and  the  values  of  x  as  abscissae, 
this  curve  will  be  a  straight  line  (Fig.  30),  which  will  pass  through 
the  origin,  since  kx  is  zero  when  x  is  zero.  To  find  the  work 
done  in  increasing  the  amount  of  the  stretch  from  xl  to  #2  where 
OL  —  xl  and  ON  =#2  we  must  find  the  area 
PLNQ.  Now  this  is  equal  to  $LN(PL  +  QN). 
Hence  the  work  done  is  ^(x^—x^  (kx^  +  kx^ 
or  (\kxj-  Pa^2).  This  is  also  the  work  the 
spring  will  do  in  contracting,  since  at  each  step 
the  force  of  contraction  is  equal  to  the  force 
required  to  stretch.  If  the  initial  stretch  be  j^  3a 

zero,  0^  =  0,  and  the  work  required  to  stretch 
by  the  amount  x2  is  p£22.      While  we  have  referred  especially 
to  the  force  exerted  by  a  spiral  spring,  the  above  proof  and 
formula  evidently  apply  to  the  work  done  by  any  force  that  is 


44          MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

proportional  to  displacement.     These  we  shall  find  later  are 
very  numerous. 

57.  Power  or  Activity.  —  The  rate  at  which  an  agent  works,  or  the 
number  of  units  of  work  performed  per  unit  time,  is  called  the 
power  or  activity  of  the  agent.     In  e.g.  s.  units  the  unit  of  ac- 
tivity is  that  of  an  agent  that  does  one  erg  per  second.     As 
this  unit  is  extremely  small,  the  unit  employed  for  most  scientific 
purposes  is  107  ergs  per  second,  or  one  joule  per  second,  and  is 
called  the  watt;  a  still  larger  unit  is  the  kilowatt  which  equals  one 
thousand  watts. 

The  unit  largely  employed  for  engineering  purposes  is  the 
horse-power,  which  is  the  power  of  an  agent  that  does  550  foot 
pounds  per  second  or  33,000  foot  pounds  per  minute.  One  horse- 
power equals  745.8  watts. 

58.  Kinetic  Energy.  —  It  is  often  necessary  to  find  the  relation 
between  the  work  done  by  a  force  and  the  velocity  of  the  body  to 
which  it  is  applied,  as,  for  example,  in  considering  the  motion  of 
a  train.     As  the  simplest  case  suppose  a  body  of  mass,  moving 
with  a  velocity  VQ  at  the  beginning  of  an  interval  of  time  t,  to  be 
acted  on  by  a  single  force  F  in  the  direction  of  the  velocity,  and 
let  the  velocity  at  the  end  of  the  time  be  v  and  the  distance  tra- 
versed be  s.     Then  from  §§27  and  45 


and  Ft  =  m(v-vQ)  (2) 

Hence  Fs  =  Jwv2  —  |mv02  (3) 


One-half  the  product  of  the  mass  of  a  body  and  the  square  of  its 
velocity  is  called  the  kinetic  energy  of  the  body,  or,  briefly, 


K.  E. 

We  may,  therefore,  state  the  above  conclusion  thus: 
Work  done  on  body  =  gain  of  kinetic  energy; 

but  it  must  be  remembered  this  is  only  for  the  case  in  which  the 
force  acts  on  a  body  which  is  otherwise  free.  It  is,  however,  true 
in  all  cases,  if  F'  stands  for  the  resultant  of  all  the  forces  acting  on 
the  body. 

We  may  also  reverse  the  circumstances  and  enquire  what  work 
a  body  in  motion  can  do,  if  it  meets  an  opposing  force  and  is 


DYNAMICS  45 

brought  to  rest.  Suppose  that  it  exerts  a  constant  force  F  and 
does  work  Fs.  Then  the  opposing  force,  that  is  the  force  applied 
to  the  body,  will  be  —F.  Making  this  change  in  (2)  we  get 

Fs  =  $mvo*  —  %mv* 

Thus  the  work  done  by  the  body  against  the  resistance  is  equal  to 
the  loss  of  kinetic  energy  of  the  body. 

If  the  motion  continue  until  the  body  is  brought  to  rest,  v  will 
then  be  zero,  and  we  shall  have  the  result  that  the  initial  kinetic 
energy  of  the  body  is  the  work  it  can  do  before  it  is  brought  to  rest. 

It  should  be  noticed  that,  in  the  above,  v  and  v0  stand  for  the 
magnitudes  of  the  respective  velocities,  i.e.,  the  speeds  (§16). 
The  kinetic  energy  of  a  body  depends  on  the  square  of  the  magni- 
tude of  its  velocity  and  is  the  same  no  matter  what  the  direction 
of  motion,  that  is,  kinetic  energy  is  a  scalar  quantity;  to  the 
kinetic  energy  of  one  body  we  may  add  the  kinetic  energy  of 
another  body  and  the  sum  will  be  the  total  kinetic  energy  of  both 
bodies. 

Since  a  force  does  no  work  when  it  is  always  at  right  angles  to 
the  direction  of  motion,  it  follows  that,  when  a  body  is  acted  on 
by  a  single  force  at  right  angles  to  the  direction  of  motion,  the 
kinetic  energy  of  the  body  remains  constant.  Thus,  when  a  body 
rotates  in  a  circle  under  the  action  of  a  single  force  directed 
toward  the  center,  the  force  does  no  work  and  the  kinetic  energy 
of  the  body  is  constant. 

Kinetic  energy  and  work  are  equivalent  quantities;  hence  the 
units  of  kinetic  energy  are  the  same  as  the  units  of  work. 

59.  Kinetic  Energy  and  Gravity.  —  The  force  of  gravity  on  a 
body  is,  for  small  distances  above  the  surface  of  the  earth,  a  con- 
stant force.  If  a  body  at  a  height  H  above  the  earth's  surface  has 
a  velocity  v  vertically  downward,  when  it  has  fallen  so  that  its 
distance  above  the  surface  is  h}  gravity  will  have  done  an  amount 
of  work  mg(H-h);  and,  if  the  velocity  of  the  body  be  then  V, 
its  kinetic  energy  will  have  increased  from  Jrav2  to  JwF3.  Hence 


If  on  the  other  hand  the  body  be  projected  upward  with  a  ve- 
locity V  from  a  height  h,  it  will  be  opposed  by  the  force  mg,  and  the 
work  it  will  do  against  gravity,  in  rising  to  a  height  H,  will  be 


46          MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

mg(H-h).  If  its  velocity  at  the  height  H  be  v,  its  loss  of 
kinetic  energy  will  be  (JraV2—  \mv*).  Equating  the  work  done 
against  gravity  to  the  loss  of  kinetic  energy  we  get  the  same 
equation  as  above. 

In  the  preceding  we  have  supposed  the  motion  to  be  vertical; 
but  the  result  will  be  unchanged  if  the  motion  is  not  vertical, 
provided  no  force  except  gravity  act  on  the  body  in  the  direction 
of  its  motion.  Any  force  perpendicular  to  the  motion  will  do  no 
work  and  cause  no  change  of  kinetic  energy.  Suppose,  for  ex- 
ample, the  body  slides  down  a  smooth  plane  through  a  distance  s 
along  the  plane.  Now,  we  have  already  shown  that  it  acquires  the 
same  velocity  as  if  it  fell  vertically  a  distance  equal  to  the  height 
of  the  plane  (§35).  Then,  if  H  be  the  height  of  the  top  of  the 
plane  and  h  that  of  the  bottom,  the  general  equation  above  will 
still  hold.  The  same  is  true  if  the  descent  is  along  a  smooth 
curve;  for  a  curve  may  be  regarded  as  made  up  of  very  short 
straight  parts  to  each  of  which  the  principle  stated  will  apply. 
These  results  are  now  readily  understood  by  considering  the  work 
performed  by  gravity.  For  the  total  amount  of  motion  in  the 
direction  of  the  whole  force  of  gravity  is  (H  —  h).  Thus  the 
gain  of  kinetic  energy  in  the  descent  from  the  higher  level  to  the 
lower  must  be  the  same  as  if  the  fall  were  vertical. 

60.  Kinetic  Energy  and  Elasticity.  —  When  a  body  is  acted  on  by 
the  force  due  to  a  stretched  spiral  spring,  the  spring  will  do  work 
on  the  body  if  the  spring  is  contracting,  and  the  body  will  do  work 
against  the  force  of  the  spring  if  it  is  moving  so  as  to  stretch 
the  spring  further.  Let  us  first  suppose  that  the  body  is  moving 
toward  the  spring  with  a  velocity  v,  the  spring  being  at  that 
moment  stretched  to  an  amount  X  beyond  its  normal  or  un- 
stretched  length.  While  the  spring  is  contracting  the  velocity  of 
the  body  will  be  constantly  increasing.  Let  the  velocity  be  V 
when  the  spring  has  contracted  so  that  its  stretch  is  decreased  to  x. 
In  this  time  (§56)  the  spring  will  have  done  an  amount  of  work 
(%kX*—  %kx2)  and,  since  this  must  equal  the  increase  of  kinetic 
energy  of  the  body, 


We  may  also  suppose  the  case  reversed,  that  is,  we  may  suppose 
the  body  to  be  moving  away  from  the  spring  with  a  velocity  V 


DYNAMICS  47 

when  the  stretch  of  the  spring  is  x.  Then  the  velocity  of  the 
body  will  decrease;  and,  if  it  be  v  when  the  stretch  of  the  spring 
is  X,  work  (%kX*  —  %kx*)  will  have  been  done  against  the  spring 
and  the  decrease  of  the  kinetic  energy  will  be  (iraV2— Jrat;3). 
Equating  these  we  get  the  same  equation  as  before. 

61.  Potential  Energy. — We  shall  now  consider  the  two  illus- 
trations just  given  from  another  point  of  view.  In  the  case  of  a 
body  projected  vertically  upward,  there  is  a  loss  of  kinetic  energy 
equal  to  mg  multiplied  by  the  height  of  ascent;  and,  if  the  body 
be  allowed  to  descend  again,  the  same  amount  of  work  will  be 
performed  by  gravity  and  the  body  will  regain  its  lost  kinetic 
energy.  Thus  at  the  higher  level  the  body  (or  rather  the  body 
and  the  earth  regarded  as  one  system)  has  an  advantage  of  posi- 
tion that  is  equivalent  to  a  certain  amount  of  kinetic  energy  lost, 
and  this  advantage  of  position  is  measured  by  mg(H—  h).  This, 
since  it  is  equivalent  to  a  certain  amount  of  kinetic  energy,  is 
called  potential  energy.  Thus  it  follows  that  the  sum  of  the  kin- 
etic energy  and  the  potential  energy  is  a  constant,  a  fact  brought 
out  more  clearly  by  writing  the  equation  of  §59  thus: 

%mV2  +  mgh  —  %mv*  +  mgH 

Here  mgh  is  the  increase  of  the  potential  energy  when  the  body 
is  raised  from  the  arbitrary  zero  level  (e.g.,  sea-level)  from  which 
h  is  measured  to  the  height  h,  and  a  similar  statement  applies  to 
mgH.  When  the  body  is  at  the  zero  level,  it  and  the  earth  still 
possess  potential  energy,  since  work  could  be  obtained  by  allow- 
ing the  body  to  fall  down  a  vertical  shaft. 

Again,  in  the  case  of  the  work  done  against  a  spring  by  a 
moving  body,  there  is  a  decrease  of  kinetic  energy,  and  this 
decrease  is  equal  to  the  work  done  against  the  spring.  If  the 
motion  be  reversed,  the  lost  kinetic  energy  will  be  regained. 
Thus  when  the  stretch  of  the  spring  increases  from  x  to  X  the 
spring  acquires  a  capacity  for  doing  work  of  the  amount 
(%kX2—  i&z2),  equal  to  the  kinetic  energy  lost  by  the  body;  and 
the  spring  yields  up  this  capacity  for  doing  work  in  restoring 
the  kinetic  energy  of  the  body.  Writing  the  equation  of  §60  in 
the  form 


48          MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

we  see  that  the  sum  of  the  kinetic  energy  of  the  body  and  the 
work  the  spring  can  do  in  contracting  to  its  unstretched  length  is 
a  constant.  The  work  the  spring  can  do  in  contracting  to  its 
unstretched  length  is  the  potential  energy  of  the  spring. 

In  the  case  of  a  body  separated  from  the  earth  the  potential 
energy  of  the  body  and  the  earth  depends  on  their  relative  posi- 
tion, and  in  the  case  of  the  energy  of  the  spring  the  potential 
energy  depends  on  the  relative  positions  of  the  parts  of  the 
spring.  Hence  we  may  say  that  potential  energy  is  the  capacity 
a  body  or  system  of  bodies  has  for  doing  work  in  virtue  of  the  relative 
positions  of  its  parts. 

In  the  case  of  potential  energy  we  cannot  give  any  universal 
formula  by  which  it  can  be  calculated  as  we  can  in  the  case  of 
kinetic  energy.  In  each  case  of  potential  energy  we  must  calcu- 
late how  much  work  the  body  or  system  can  do  in  passing  from 
one  state  to  another,  and  take  this  as  the  difference  of  the  poten- 
tial energy  of  the  body  or  system  in  the  two  states.  For  any  one 
particular  case  of  potential  energy  we  may  deduce  a  special 
expression  for  its  amount,  such  as  those  given  above  for  gravity 
and  elasticity. 

From  the  statements  made  in  §§58-61,  it  is  evident  that  we 
may  define  energy,  of  either  kind,  as  the  capacity  for  doing  work. 

62.  Interchanges  of  Kinetic  and  Potential  Energy. — We  have 
considered  somewhat  fully  two  cases  of  the  interchange  of  kinetic 
and  potential  energy,  namely,  those  of  gravity  and  elasticity,  be- 
cause these  are  typical  and  are  easily  worked  out  by  elementary 
methods.  Such  interchanges  are  common  in  nature  and  in 
industry  and  a  few  may  be  briefly  stated. 

(a)  Change  from  Kinetic  to  Potential. — When  a  block  of  wood 
is  split  by  a  wedge  or  axe,  the  axe  or  sledge  hammer  loses  kinetic 
energy  and  potential  energy  of  separation  of  the  particles  of 
wood  is  produced. 

As  the  distance  of  the  earth  from  the  sun  increases  from  mid- 
winter to  midsummer,  the  speed  of  motion  and  the  kinetic  energy 
decrease  and  the  potential  energy  of  separation  increases. 
JS'  (b)  Change  from  Potential  to  Kinetic. — A  clock  w-eight  or  watch- 
spring  when  wound  up  has  potential  energy,  and  this  changes  to 
kinetic  energy  of  the  pendulum  or  balance  wheel,  which  would 
otherwise  come  to  rest. 


DYNAMICS  4.9 

A  bent  bow  has  potential  energy  due  to  the  change  of  position 
of  the  particles  of  the  bow  and  the  forces  between  them.  As  it 
unbends  it  loses  this  potential  energy  and  the  arrow  gains  kinetic 
energy. 

Water  in  a  lake  or  above  a  dam  has  potential  energy;  when  al- 
lowed to  escape  to  a  lower  level,  it  loses  part  of  its  potential  energy 
and  either  gains  kinetic  energy  itself  or,  if  it  acts  on  a  water  wheel 
or  turbine,  it  imparts  kinetic  energy  to  the  latter. 

(c)  Periodic  Interchanges. — In  any  case  of  vibration  energy 
continually  changes  from  the  kinetic  to  the  potential  form  and 
back  again.  Thus,  in  the  vibration  of  a  pendulum,  at  the  bottom 
of  the  arc  of  vibration  the  potential  energy  is  at  a  minimum  and 
the  kinetic  energy  is  at  a  maximum,  while,  at  the  end  of  the  arc 
of  vibration,  the  kinetic  energy  is  zero  and  the  potential  energy 
has  increased  to  a  maximum.  Similar  statements  apply  to  the 
vibration  of  a  tuning  fork,  a  violin  string,  a  body  attached  to  the 
end  of  a  wire  and  vibrating  torsionally,  the  oscillations  of  the 
balance-wheel  of  a  watch,  and  so  on. 

63.  Two  Kinds  of  Forces. — In  the  preceding  we  have  seen  that, 
when  the  forces  acting  between  bodies  are  forces  of  gravity  or 
forces  of  elasticity,  their  action  leaves  the  total  kinetic  and 
potential  energy  of  the  bodies  unchanged,  or,  as  it  is  usually 
stated,  when  only  such  forces  act  the  total  kinetic  and  potential 
energy  is  conserved.  Forces  whose  action  between  bodies  does 
not  cause  a  change  of  the  total  kinetic  and  potential  energy  of 
the  bodies  are  called  conservative  forces ,  and  any  system  of  bodies 
between  which  the  forces  are  wholly  conservative  is  called  a 
conservative  system. 

In  contrast  with  these  conservative  forces  stands  such  a  force 
as  friction.  A  moving  body  opposed  by  friction  loses  kinetic 
energy  as  its  velocity  decreases,  but  it  does  not  at  the  same  time 
gain  potential  energy  to  an  equivalent  extent.  Thus,  a  body 
started  up  a  rough  inclined  plane  with  a  certain  velocity  will 
not  reach  as  high  a  level  as  it  would  reach  if  the  plane  were 
smooth,  and  it  will  not  have  as  much  potential  energy  when  it 
reaches  its  highest  point.  Moreover,  its  descent  will  be  further 
opposed  by  friction,  and  its  store  of  kinetic  and  potential  energy 
will  thereby  be  further  reduced.  Friction,  then,  is  a  non-con- 


50  MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

servative  force  since,  when  in  action,  it  causes  a  permanent 
decrease  of  the  kinetic  and  potential  energy  of  a  system. 

The  reason  why  such  a  force  as  gravity  has  no  effect  on  the 
sum  total  of  kinetic  and  potential  energy  is  easily  seen.  At  a 
certain  distance  of  a  body  from  the  earth  the  force  between  the 
two  depends  only  on  their  distance  apart,  and  is  independent  of 
the  way  in  which  they  are  moving.  Hence,  when  they  are 
moving  away  from  each  other  and  are  a  certain  distance  apart, 
they  are  losing  kinetic  energy  at  a  rate  exactly  equal  to  the  rate 
at  which  they  regain  kinetic  energy  when,  at  the  same  distance 
of  separation,  they  are  moving  toward  one  another.  Thus 
forces  of  gravity  between  bodies  depend  only  on  the  relative 
positions  of  the  bodies.  The  same  is  true  of  the  forces  between 
the  parts  of  an  elastic  spring,  and  this  accounts  for  the  fact  that 
such  forces  of  elasticity  are  also  conservative;  in  fact  it  is  the 
fundamental  characteristic  of  all  conservative  forces.  But  a 
non-conservative  force,  such  as  friction,  depends  on  the  way  in 
which  a  body  or  a  system  of  bodies  is  moving;  it  is  always  opposed 
to  the  direction  of  relative  motion  of  bodies  in  contact;  hence  it 
causes  a  diminution  of  the  kinetic  energy  of  the  bodies  in  which- 
ever direction  motion  is  taking  place. 

64.  The  Conservation  of  Mechanical  Energy. — We  have  seen 
in  the  preceding  that  under  certain  conditions  the  total  kinetic  and 
potential  energy  of  a  system  is  constant  or  is  conserved.  The  con- 
ditions referred  to  are  two,  (1)  the  system  must  not  receive 
energy  from  or  give  energy  to  any  outside  bodies,  (2)  the  forces 
between  the  parts  of  the  system  must  be  wholly  conservative. 
In  reality  no  system  wholly  satisfies  these  conditions.  No  system 
is  wholly  isolated  in  the  sense  implied  in  the  first  condition;  and 
non-conservative  forces,  such  as  friction,  are  never  quite  absent. 
But  in  many  cases  these  conditions  are  very  nearly  satisfied. 
The  solar  system,  consisting  of  the  sun,  planets,  and  moons,  is 
practically  isolated;  and,  while  there  are  internal  frictional  forces 
such  as  those  of  the  tides,  the  work  they  do  is  so  small  compared 
with  the  total  energy  of  the  system,  that  their  effects  in  reducing 
the  kinetic  energy  of  the  whole  have  not  yet  been  detected  with 
certainty.  Again,  the  system  consisting  of  the  earth  and  a  body 
vibrating  as  a  pendulum  in  a  vacuum  is  practically  an  isolated 
system  free  from  frictional  forces,  and  the  total  kinetic  and 


KINEMATICS  OF  RIGID  BODIES  51 

potential  energy  is  very  nearly  constant;  the  same  is  true  of  a 
heavy  body  attached  to  a  spring  and  vibrating  in  a  vacuum. 
When,  as  in  cases  like  these,  the  conditions  are  sufficiently  nearly 
satisfied,  the  principle  of  the  constancy  of  kinetic  and  potential 
energy  will  often  lead  to  valuable  results. 

In  an  isolated  system  in  which  there  are  non-conservative 
forces,  such  as  friction,  energy  is  expended  in  doing  work  against 
these  forces;  and  if,  to  the  sum  of  the  kinetic  and  potential 
energy,  we  add  the  work  done  against  non-conservative  forces, 
the  sum  will  be  constant.  But  what  becomes  of  the  energy  so 
expended?  For  long  it  was  supposed  to  be  wholly  lost.  It  was, 
of  course,  known  that  heat  was  produced  when  work  was  done 
against  friction;  but  heat  was  supposed  to  be  a  form  of  matter. 
But  about  1840  the  view  was  advanced  that  heat,  instead  of 
being  a  form  of  matter,  is  a  form  of  energy  as  this  word  is  now 
defined,  and  this  led  to  the  discovery  of  the  Law  of  Conservation 
of  Energy,  which  is  treated  fully  later  under  "  Heat." 

KINEMATICS  OF  RIGID  BODIES 
ROTATION 

65.  Angular  Displacements. — In  §§9-35,  we  studied  the  motion 
of   translation   of    a  point,  as  a  preliminary  to  the  study  of 
the  effect  of  forces  on  the  motion  of  particles  and  of  bodies  moving 
without  rotation.     We  shall  now  consider  the  motion  of  bodies 
in  rotation,  as  a  preliminary  to  studying  the  effects  of  forces  on 
the  motion  of  rotation  of  bodies. 

The  motion  of  a  body  is  one  of  rotation  when  each  point  in  the 
body  moves  in  a  circle  the  center  of  which  is  on  a  straight  line 
called  the  axis  of  rotation.  All  points  in  the  body  turn  in  any 
time  through  equal  angles  and  the  angle  described  in  any  time  is 
called  the  angular  displacement  of  the  body  in  that  time.  Its 
magnitude  maybe  stated  in  degrees  or  in  radians  (1  radian  =  57.3° 
approx.),  but  the  latter  method  is  in  many  ways  the  more  con- 
venient for  the  present  purposes. 

66.  Angular  Velocity. — The  rate  of  rotation  of  a  body  is  called 
its  angular  velocity.     When  the  angular  displacements  of  a  body 
in  all  equal  times  are  equal,  the  velocity  is  a  constant  angular 
velocity,  and  the  magnitude  of  the  angular  velocity  is  the  angle 


52          MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

through  which  the  body  turns  in  unit  time.  If  the  angle  is  reckoned 
in  radians  and  the  second  is  taken  as  unit 'of  time,  the  magnitude 
of  the  angular  velocity  is  the  number  of  radians  described  in  one 
second.  The  unit  of  angular  velocity  is  one  radian  per  second. 

If  the  velocity  is  not  constant,  as,  for  example,  when  a  fly- 
wheel is  being  set  in  motion  or  stopped,  the  angular  velocity  or 
rate  of  angular  displacement  is  defined  in  the  same  way  as  in  the 
analogous  case  of  variable  linear  velocity  (§19),  that  is  to  say, 
we  must  take  the  average  angular  velocity  in  a  short  time  and 
then  suppose  this  time  indefinitely  decreased,  so  that  the  average 
angular  velocity  approaches  a  limiting  value,  which  is  the  in- 
stantaneous angular  velocity. 

67.  Angular  Acceleration. — The  rate  of  increase  of  the  angular 
velocity  of  a  body  is  called  its  angular  acceleration.  When  the 
angular  velocity  increases  by  equal  amounts  in  equal  times,  the 
angular  acceleration  is  constant  and  its  magnitude  is  the  increase 
of  angular  velocity  in  unit  time.  If  we  denote  the  angular  accel- 
eration by  a,  the  increase  of  angular  velocity  in  each  second  is 
a  and  the  increase  in  t  seconds  is  at  Hence,  if  at  the  beginning 
of  an  interval  of  time  t  the  angular  velocity  is  (u0  and  at  the 
end  of  the  interval  it  is  o>, 

(t)=a)9  +  cd  (1) 

In  this  time  the  body  has  turned  through  a  certain  angle  say  <j>. 
To  find  the  magnitude  of  (/>  we  may  represent  the  varying  values 
of  the  angular  velocity  by  means  of  a  curve  of  angular  velocity, 
as  we  did  in  the  similar  case  of  a  varying  linear  velocity  (§27), 
and  the  area  of  the  diagram  will  represent  the  angle  <£.  The  two 
diagrams  would  have  precisely  similar  properties,  the  only  differ- 
ence being  that  in  one  case  we  would  speak  of  linear  displace- 
ment, s,  linear  velocity,  v,  linear  acceleration,  a,  while  in  the  other 
case  we  would  speak  of  angular  displacement,  <£,  angular  velocity, 
a,  and  angular  acceleration,  a.  Hence,  when  the  angular  ac- 
celeration is  constant,  the  formula  for  <£,  which  must  be  pre- 
cisely similar  to  (2)  of  §27,  is 

0=:<Me«  +  Jarfa  (2) 

By  elimination  of  t  between  (1)  and  (2)    we  get 

(3) 


KINEMATICS  OF  RIGID  BODIES  53 

68.  Angular  Velocity  and  Linear  Velocity. — When  a  point  re- 
volves at  a  constant  rate  in  a  circle,  its  motion 

may  be  described  either  by  means  of  its  angular 
velocity,  wt  or  by  its  linear  velocity,  v,  along 
the  tangent,  and  there  is  a  simple  relation  be- 
tween the  two.  Let  the  radius  of  the  circle  be 
r  and  let  the  time  in  which  the  point  moves 
from  P  to  Q  be  t.  Denoting  the  length  of 
the  arc  PQ  by  s  and  the  angle  POQ  by  <£, 
we  have,  from  the  definitions  of  linear  and  of  FIO.  31. 

angular  velocity, 

S—Vt.  (f>=0)t. 

Now  in  radian  measurement  <f>  =  s/r.  Substituting  in  this  the 
above  values  of  s  and  <f>,  we  get 

a)  =  v/r 

Thus  the  relation  between  angular  velocity  and  linear  velocity 
when  a  point  rotates  in  a  circle  is  the  same  as  the  relation  between 
an  angle  and  the  arc  which  it  subtends. 

The  above  relation  is  important.  More  briefly  stated,  the 
proof  of  it  amounts  to  this:  v  is  the  length  of  arc  described  per 
second;  hence  v/r  is  the  angle  described  per  second  in  radian 
measurement,  that  is  the  angular  velocity. 

When  a  point  describes  a  circle  with  variable  speed,  the  above 
relation  holds  true,  with  the  understanding  that  a)  and  v  are  the 
instantaneous  values  of  the  angular  and  the  linear  velocity 
respectively.  The  proof  is  the  same  as  above,  t  being  taken  as  a 
very  short  interval. 

When  a  body  rotates  about  an  axis  with  angular  velocity  o>,  a 
point  in  the  body  describes  a  circle  of  radius  r,  and  r  is  different 
for  points  at  different  distances  from  the  axis.  If  r  and  /  are 
the  respective  distances  of  two  points  from  the  axis  and  v  and  t/ 
their  respective  linear  velocities,  v  =  ra>  and  t?'=r/ft>.  Hence 
v:v'::r:r'. 

69.  Instantaneous  Axes  of  Rotation. — When  the  axis  about 
which  a  body  rotates  varies  from  moment  to  moment,  the  above 
relation  is  true  of  the  values  of  o>,  v,  and  r  at  any  moment.    For 
example  the  wheel  of  a  moving  wagon  or  bicycle  is  always  in  con- 
tact with  the  road  and  the  point  of  contact  is  at  any  moment 


54          MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

the  point  about  which  the  whole  wheel  is  at  that  moment  rotat- 
ing. Now  the  top  of  the  wheel  is  twice  as  far  from  the  ground  as 
the  center  of  the  hub  and  must,  therefore,  have  twice  as  great  a 
linear  velocity. 

70.  Angular  Acceleration  and  Linear  Acceleration. — When  a 
point  revolves  in  a  circle  (Fig.  32)  with  changing  angular  velocity, 
it  has  an  angular  acceleration,  say  a.  The  speed  of  the  point  along 
the  tangent  increases  with  an  acceleration,  say  a.  The  same  rela- 
tion holds  between  a  and  a  as  between  v  and  oj  (§68).  For,  if  w  is 


;\ 


Fio.  32.  Fia    33. 

the  angular  velocity  at  the  beginning  of  a  short  time  t  and  v  the 
linear  speed  at  this  time,  v  =  a)r;  at  the  end  of  the  time  t  the 
angular  velocity  is  (co+at)  and  the  linear  speed  is  (v  +  at). 
Hence  (v+at)=r(a)  +  at).  Subtracting  the  former  equation 
from  the  latter  and  cancelling  t,  we  have 

o=ra 

More  briefly  stated,  a  is  the  added  linear  speed  per  unit  time  and 
a  the  added  angular  velocity  per  unit  time,  and  the  relation  be- 
tween angular  velocity  and  linear  speed  must  hold  true  of  these 
increases. 

It  should  be  carefully  noted  that  a  here  means  the  rate  of 
change  of  speed  along  the  tangent.  Since  the  direction  of  the 
velocity  is  also  changing  this  cannot  be  the  only  acceleration.  In 
fact,  as  we  have  already  seen  (§32),  there  is  in  all  cases  of  motion 
in  a  curve  a  linear  acceleration  toward  the  center  equal  to  v2/r, 
or,  as  we  may  now  write  it,  o>2r,  since  v=rtt). 

The  above  relations,  which  are  very  important,  are  summarized 
in  Fig.  33. 

71.  Graphical  Representation  of  Angular  Quantities. — An  angu- 
lar displacement  is  of  a  certain  magnitude  and  is  about  a  certain 


KINEMATICS  OF  RIGID  BODIES  55 

axis.  Given  the  axis,  the  direction  of  rotation  around  it,  and 
the  magnitude  of  the  angular  displacement,  we  know  everything 
about  it.  Now  all  these  can  be  represented  graphically  by  a 
length  marked  off  on  the  axis  so  as  to  represent  to  some  scale  (e.g., 
a  cm.  per  radian)  the  magnitude  of  the  angular  displacement. 
There  must  also  be  some  agreement  as  to  which  direction  along 
the  axis  shall  represent  a  certain  direction  of  rotation  about  the 
axis.  The  rule  usually  adopted  for  this  purpose 
is  called  the  "right-handed  screw  rule,"  namely, 
let  the  direction  (along  the  axis)  of  the  line  that 
represents  an  angular  displacement  be  related  to 
the  direction  of  the  rotation  as  the  direction  of 
translation  is  to  the  direction  of  rotation  of  an 
ordinary  (right-handed)  screw.  For  example,  a  FlQ  34_Arota. 
line  to  represent  the  angular  displacement  of  the  tion  indicated  by 
earth  in  24  hours  due  to  its  rotation  about  its 
axis  would  be  drawn  from  the  center  toward  the 
N.  pole.  Two  lines  to  represent  the  angular  displacements  of 
the  hands  of  a  watch  in  one  hour  would  be  drawn  through  the 
center  of  the  face  toward  the  back  and  the  one  for  the  minute 
hand  would  be  twelve  times  as  long  as  the  one  for  the  hour  hand. 
A  line  to  represent  an  angular  velocity  would  be  laid  off  on  the 
axis  of  rotation  according  to  the  above  rule,  and  a  line  to  repre- 
sent an  angular  acceleration  would  be  drawn  in  the  same  way 

A  directed  line  that  represents  according  to  the  above  agreement  an 
angular  displacement  is  a  vector,  since  it  has  both  magnitude  and  direction; 
but  it  differs  from  vectors  that  represent  linear  displacements  in  the  fact 
that  it  must  in  any  diagram  be  located  on  a  certain  line,  namely,  the  line 
that  stands  for  the  axis  of  rotation.  Such  a  vector  is  therefore  called  a 
localized  vector  or  rotor.  Two  parallel  and  equal  vectors  of  this  kind,  not  in 
the  same  line,  do  not  represent  the  same  angular  displacement,  since  the 
rotations  they  represent  are  about  different  axes. 

72.  Addition  of  Angular  Velocities  and  Accelerations  about 
Intersecting  Axes. — A  body  may  have  two  or  more  simultaneous 
angular  velocities.  For  example,  suppose  a  bicycle  wheel  while 
rotating  about  its  axis  is  mounted  on  a  horizontal  platform  which 
is  kept  in  rotation  about  a  vertical  axis.  At  any  moment  the 
wheel  has  two  component  angular  velocities  about  intersecting 
axes.  Each  may  be  represented  by  a  vector  drawn  from  the 
centei  according  to  the  rule  stated  in  §71.  We  may  then  add 


56          MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

these  two  vectors  by  the  parallelogram  method,  and  the  diagonal 
will  represent  in  magnitude  and  direction  the  resultant  angular 
velocity  at  the  moment  in  question.  (This  is  fully  proven  in 
advanced  treatises.) 

Since  angular  accelerations  are  increments  of  angular  velocities 
per  unit  time,  we  may  add  them  as  we  add  angular  velocities. 

It  follows  from  the  above  that  angular  velocities  about  inter- 
secting axes  may  be  compounded  and  resolved  by  the  methods 
applicable  to  linear  velocities  (§§23-25),  and  a  similar  statement 
holds  true  for  angular  accelerations  about  intersecting  axes. 

DYNAMICS   OF  RIGID  BODIES 
CENTER  OF  MASS 

73.  General  Description  of  Center  of  Mass. — When  the  motion 
of  a  rigid  body  is  one  of  translation  without  rotation,  all  points  in 
the  body  move  in  exactly  the  same  way,  and,  in  describing  or 
calculating  the  motion,  any  point  in  the  body  may  be  taken  as 
representing  the  whole  body.  When  the  motion  is  one  of  trans- 
lation combined  with  rotation,  different  points  in  the  body  move 
differently  and  there  is  no  one  point  the  motion  of  which  com- 
pletely represents  the  motion  of  the  whole  body.  There  is,  how- 
ever, in  any  body  one  particular  point  which,  for  many  purposes, 
may  be  taken  as  representing  the  body,  so  that  for  these  purposes 
the  body  may  be  regarded  as  concentrated  to  a  particle  at  that 
point.  This  point,  which  we  shall  define  presently,  is  the  center 
of  mass  of  the  body.  For  instance,  let  a  uniform  circular  disk  be 
tossed  into  the  air;  it  will  be  seen  that  the  center  of  the  disk 
moves  like  a  particle  either  in  a  straight  line  or  in  a  parabola, 
while  other  points  in  the  disk  rotate  around  it.  If  the  disk  is 
loaded  with  lead  on  one  side  it  will  be  some  other  point,  not  the 
geometrical  center,  that  will  show  this  property. 

If  a  body  wholly  free  were  struck  a  blow  at  random,  it  would 
start  with  both  translation  and  rotation;  but  if  the  blow  were 
applied  at  the  center  of  mass  or  in  a  line  through  the  center  of 
mass,  the  motion  would  be  one  of  translation  without  rotation. 

The  center  of  mass  is  thus  seen  to  be  a  point  of  great  im- 
portance in  describing  or  calculating  the  whole  motion  of  a  body. 
In  what  follows  we  shall  define  the  center  of  mass  and  show  how 


DYNAMICS  OF  RIGID  BODIES 


57 


its  position  may  be  calculated.     Then  from  the  definition  we 
shall  deduce  the  above  and  other  properties. 

74.  Center  of  Mass  of  a  Number  of  Particles. — The  meaning  of 
the  center  of  mass,  in  general,  will  be  more  clearly  understood  if 
we  begin  with  some  simple  cases  that  suggest  the 
^/>;  general  definition.     (1)  Two  Particles.     Let  the 
£7/        particles  be  m1  at  Pj  and  m2  at  P2.     Let  Cl  be  a 
point  that  divides  Pft  inversely  as  the  masses 
of  the  particles,  that  is,  such  that 


Fia.  35.  _  _ 

Cl  is  the  center  of  mass  of  mt  and  mr 
(2)   Three  Particles.     Let  the  particles  be  ml  and 
m2  as  above  and  ras  at  P8.     Suppose  m^  and  m2 
replaced  by  (m1  +  m2)  at  C  and  let  C2  be  a  point  in 
CiP3  such  that  (Fig.  36) 


C2  is  the  center  of  mass  of  mlf  m2  and  m3.  FIO.  36. 

(3)  Any  Number  of  Particles.      Proceeding   as 
above  we  get  the  center  of  mass,  C,  of  any  number  of  particles, 
and  the  same  will  apply  to  a  body  of  any  form,  since  it  may  be 
divided  up  into  a  large  number  of  small  parts. 

We  shall  show  in  the  next  section  that  the  point  to  which 
such  a  process  leads  is  independent  of  the  order  in  which  the 
particles  are  taken. 

75.  Distance  of  Center  of  Mass  from  a  Plane. — Let  EF  (Fig. 
37)  be  the  line  in  which  any  plane  is  cut  by  a  perpendicular 


E 
L 

N 

M 

r 

di 

D,    Q2C/ 

d2       / 

P™* 

\ 


Fia.  37. 


Fia.  38. 


plane  through  PtP2  of  the  last  section.     Draw  P4L,  P2M,  C^N 
perpendicular  to  EF  and  denote  their  lengths  by  dt,  c?2  and  Dit 


58          MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

respectively.  Draw  PiQi  and  PzQz  perpendicular  to  CiN. 
Since  CiQi  and  C2Q2  are  the  projections  of  CJPi  and  CiP2  it  is 
readily  seen  from  the  equation  in  §74  (1)  that 


or 


If  we  should  proceed  to  apply  the  same  method  to  (ml  +  m2)  at 
Ct  and  m8  at  P,  (Fig.  38)  we  would,  it  is  evident,  get  a  similar 
result.  Hence 

(Wj+Wj-f  ra8)D2=  (ml+m2)Dl 

Hence,  substituting  from  the  above, 


By  extending  the  same  method  to  any  number  of  particles  we 
shall  evidently  obtain  the  general  formula 


It  is  evident  that  this  result  will  not  be  altered  if  the  order  in 
which  the  various  particles  are  taken  be  altered  in  any  way, 
and  that  it  is  true  whatever  plane  of  reference  is  chosen. 

76.  General  Definition  of  Center  of  Mass.  —  Itm^  ra2,-  •  •  are  the 
respective  masses  of  the  particles  constituting  a  body  (or  group  of 
particles)  of  total  mass  M,  and  if  the  respective  distances  of 
these  particles  from  any  plane  are  dlt  d2,  •  •  •,  the  center  of  mass 
is  a  point  whose  distance  from  the  plane  is 


..          M 

If  in  any  case  one  or  more  of  the  distances  are  measured  on  the 
opposite  side  of  the  plane  from  the  others,  when  we  substitute 
numbers  for  the  various  distances  those  corresponding  to  one 
side  of  the  plane  must  be  given  positive  signs  and  the  others 
negative. 

If  the  plane  from  which  dlf  d2-  •  -are  measured  passes  through 
the  center  of  mass,  D  is  zero  and  in  this  case 


When  in  any  case  it  is  desired  to  find  the  position  of  the  center 
of  mass  of  a  body  by  applying  the  above  formula,  it  is  only 


DYNAMICS  OF  RIGID  BODIES  59 

necessary  to  apply  it  to  distances  from  three  planes  at  right 
angles.  Denoting  the  distances  from  one  of  them  by  X'B,  from 
a  second  by  t/'s,  from  the  third  by  z's,  we  get 

Sras     -      Itmy  limz 

»-TJT'   y=~M~'       "  M 

where  x  denotes  the  distance  of  the  center  of  mass  from  the  plane 
from  which  the  re's  are  measured,  and  similarly  for  y  and  z. 

77.  Center  of  Mass  of  a  Regular  Body. — The  center  of  mass  of 
two  equal  particles  is  at  the  middle  of  the  line 

joining  them.  A  uniform  rod  may  be  divided 
into  pairs  of  equal  particles,  the  two  in  each  pair 
being  equidistant  from  the  center  of  the  rod. 
Hence  the  center  of  mass  of  the  whole  rod  is  at 
its  middle  point.  Similar  reasoning  may  be  ap- 
plied to  any  homogeneous  body  which  has  a  Q  L  R 
geometrical  center  such  as  a  circle,  ellipse,  sphere,  ^ia.  39. 

spheroid,  parallelogram,  cube,  parallelepiped,  etc. 
The  center  of  mass  of  each  of  these  is  at  its  geometrical  center. 

When  a  body  can  be  divided  into  parts  such  that  the  center  of 
mass  of  each  is  known,  the  center  of  mass  of  the  whole  can  usually 
be  found.  A  triangle  may  be  divided  into  narrow  strips  parallel 
to  one  side;  the  center  of  mass  of  each  strip  lies  on  the  line  joining 
the  middle  of  that  side  to  the  opposite  vertex.  Hence  the  center 
of  mass  of  a  triangle  is  at  the  intersection  of  the  three  lines  which 
join  the  vertices  to  the  middle  of  the  opposite  sides.  Similar 
reasoning  shows  that  the  center  of  mass  of  a  triangular  pyramid 
is  at  the  intersection  of  the  four  lines  that  join  the  vertices  to  the 
respective  centers  of  mass  of  the  opposite  faces. 

78.  Velocity  and  Acceleration  of  the  Center  of  Mass. — Let  us 
suppose  that  the  velocity  of  each  particle  in  a  group  of  particles 
is  known.     How  can  the  velocity  of  the  center  of  mass  be  found? 
To  answer  this  it  is  sufficient  to  show  how  the  velocity  of  the 
center  of  mass  in  a  direction  perpendicular  to  each  of  three  planes 
at  right  angles  can  be  found. 

To  find  the  velocity  of  the  center  of  mass  in  a  direction  per- 
pendicular to  any  plane,  consider  the  distances  of  the  particles 
and  of  the  center  of  mass  from  that  plane.  These  are  connected 
by  the  equation  (§75) 

•  •  •  (1) 


60         MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

At  a  time  t  later  these  distances  will  have  all  changed.     Let  the 
new  values  of  the  distances  be  d/,  d2'  •  •  •  D'.     Then 

(mt+m2-f  •-)D'  =  mldl'+m2di'+---  (2) 

Subtract  each  side  of  (1)  from  the  corresponding  side  of  (2); 
divide  through  by  t  and  suppose  t  decreased  without  limit.     Then 
(D'  —  D)/t  will  become  the  velocity,  say  v,  of  the  center  of  mass; 
(df  —  d^lt  will  become  the  velocity,  say  vl}  of  ml  and  so  on. 
Hence 

2+  •  •  •  (3) 


Thus  the  velocity  of  the  center  of  mass  is  related  to  the  velocities 
of  the  separate  particles  as  the  distance  of  the  center  of  mass  from 
any  plane  is  related  to  the  distances  of  the  particles  from  that 
plane. 

We  may  now  proceed  to  apply  the  same  reasoning  to  find  the 
acceleration  of  the  center  of  mass.  Starting  with  (3)  above,  let 
us  consider  what  (3)  becomes  at  a  short  time  t  later.  We  shall 
thus  get  two  equations.  Subtracting  one  from  the  other  as  be- 
fore, dividing  by  t,  and  then  supposing  t  indefinitely  short,  we  get 

(ml+m2  +  •  •  ')a  =  mlal+m2a2+  •••  (4) 

Equation  (3)  is  readily  obtained  by  differentiating  (1)  with  reference  to 
the  time  (see  §19)  and  (4)  is  obtained  by  differentiating  (3)  (see  §31). 

79.  Acceleration  of  Center  of  Mass  due  to  External  Forces.  — 

Equation  (4)  of  the  last  section  has  a  very  important  interpreta- 
tion. The  term  m^  is,  by  the  Second  Law  of  Motion,  equal  to 
the  force  that  acts  on  ml  in  the  direction  in  which  a  is  measured, 
which,  of  course,  may  be  any  direction,  and  similarly  for  the  other 
particles.  Now  the  forces  may  be  divided  into  two  groups,  (1) 
forces  applied  from  the  outside  or  external  forces  such  as  gravity 
acting  on  the  body,  pressures  and  pulls  applied  to  the  surface  of 
the  body  and  so  on;  (2)  forces  that  the  particles  exert  on  one 
another,  that  is  internal  forces,  actions  and  reactions  between  the 
particles.  By  the  Third  Law  of  Motion  these  internal  forces 
occur  in  pairs  of  equal  and  opposite  forces,  and  the  sum  of  the 
components  of  all  of  them  in  any  direction  is  zero. 

Hence  the  right  hand  side  of  (4)  stands  for  the  sum  of  the  com- 


DYNAMICS  OF  RIGID  BODIES  61 

ponents,  in  the  direction  considered,  of  all  the  external  forces. 
Thus  if  M  be  the  whole  mass  of  the  body  or  group  of  particles, 

_  _  sum  of  components  of  external  forces 
M 

Now  by  the  Second  Law  of  Motion  this  is  the  expression  we  would 
arrive  at  if  we  asked,  "what  acceleration  would  the  center  of 
mass  of  the  body  receive  if  the  whole  mass  were  concentrated 
there  and  all  the  external  forces  were  transferred  parallel  to 
themselves  so  as  to  act  at  that  point? ' 

Hence  the  center  of  mass  of  a  body  moves  as  if  the  whole  mass  were 
concentrated  at  the  center  of  mass  and  the  forces  acting  on  the  body 
were  transferredt  with  their  directions  unchanged,  to  the  center  of 
mass. 

We  now  see  the  explanation  of  the  facts  stated  in  §73.  In  the 
case  of  a  body  tossed  into  the  air  gravity  is  the  only  external 
force,  and  the  center  of  mass  moves  as  if  all  the  mass  and  weight 
were  concentrated  there,  that  is,  it  moves  as  a  particle  would. 
Even  when  a  body  has  its  form  changed  very  abruptly  by  the 
action  of  internal  forces,  as  in  the  case  of  the  explosion  of  a  rocket, 
the  internal  forces  do  not  affect  the  motion  of  the  center  of  mass 
of  all  the  particles.  When  two  bodies  approach  and  impinge,  the 
motion  of  their  center  of  mass  is  not  affected  by  the  forces  be- 
tween the  bodies  during  impact,  and  hence  continues  unchanged 
after  the  impact.  There  are  powerful  forces  of  attraction  be- 
tween the  sun  and  the  planets  that  make  up  our  solar  system,  but 
the  center  of  mass  of  the  whole  moves  with  a  uniform  velocity 
through  space. 

80.  Translation  and  Rotation. — It  is  evident  that  to  ascertain 
the  whole  motion  of  a  body  it  is  sufficient  to  find:  (1)  The  motion 
of  translation  of  some  point  in  the  body.  (2)  The  motion  of  rota- 
tion about  that  point.  By  the  result  obtained  in  §79  we  can  find 
the  linear  acceleration  of  the  center  of  mass  from  the  magnitudes 
and  directions  of  the  forces,  without  considering  the  distances 
of  their  lines  of  action  from  the  center  of  mass. 

We  shall  now  consider  how  the  angular  acceleration  of  a  body 
can  be  calculated,  but  in  an  elementary  work  it  is  necessary  to 
confine  attention,  for  the  most  part,  to  the  rotation  of  rigid  bodies 
mounted  on  fixed  axes. 


62          MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

MOMENTS  OF  FORCE  AND  MOMENTS  OF  INERTIA 

81.  When  a  rigid  body  is  mounted  on  a  fixed  axis  (e.g.  a 
grindstone  or  fly-wheel)  the  only  motion  that  a  force  applied  to  it 
can  produce  is  one  of  rotation  about  the  axis.  To  find  the  magni- 
tude of  the  effect  we  must  consider,  not  only  the  magnitude  and 
the  direction  of  the  force,  but  also  the  distance  of  its  line  of  action 
from  the  axis.  For  it  is  a  matter  of  common  experience  that  a 
force  can  be  most  effectively  applied  to  set  a  large  body  into 
rotation  when  it  is  applied  as  far  from  the  axis  as  possible. 

On  the  other  hand,  the  inertia  resistance  which  the  force 
encounters  depends  on  something  more  than  the  mass  of  the 
body.  For  it  is  also  well  known  that  the  farther,  on  the  whole, 
the  mass  of  the  body  is  from  the  axis,  as,  for  example,  in  the  case 
of  a  fly-wheel  with  a  heavy  rim  and  light  spokes,  the  harder  it  is 
to  set  it  into  rotation  or  to  stop  it. 

We  are  thus  led  to  consider  moments   of 
force  and  moments  of  inertia. 

82.  The  Moment  of  a  Force. — Consider  a 
body  mounted  on  a  fixed  axis,  A,  perpendic- 
ular to  the  plane  of  the  paper.     Let  a  force, 
FIG.  40.  /»  act  on  tne  b°dy,  the  line  of  action  of  the 

force  being  in  a  plane  perpendicular  to  the 
axis,  and  let  the  perpendicular  distance  of  the  line  of  action  from 
the  axis  be  p.  The  product  fp  is  called  the  moment  of  /  about 
A.  It  depends  on  the  magnitude,  direction,  and  line  of  action 
of  the  force;  but  it  does  not  depend  on  the  particular  point  in 
the  line  of  action  at  which  /  is  applied.  A  moment  of  force  is 
also  called  a  torque. 

The  above  is  not  a  general  definition  of  the  moment  of  a  force, 
for  we  have  supposed  the  line  of  action  of  the  force  to  be  in  a  plane 
perpendicular  to  the  axis.  To  find  the  moment  of  a  force,  F,  in 
any  direction,  we  must  first  resolve  F  into  a  component  parallel 
to  the  axis  and  a  component,  say  /,  perpendicular  to  the  axis. 
The  former  cannot  produce  motion  about  the  axis,  since  it  is 
parallel  to  the  axis.  The  latter  component,  /,  is  the  effective 
component. 

The  moment  of  a  force  about  an  axis  is  the  product  of  the  com- 
ponent of  the  force  perpendicular  to  the  axis  (the  other  component 


DYNAMICS  OF  RIGID  BODIES  63 

being  parallel  to  the  axis)  by  the  perpendicular  distance  of  this 
component  from  the  axis. 

Since  one  direction  of  rotation  about  an  axis  is  taken  as  posi- 
tive, the  other  being  taken  as  negative,  moments  of  forces  are 
considered  as  positive  or  negative  according  to  the  directions  in 
which  they  tend  to  produce  rotation. 

A  moment  of  force,  although  it  is  the  product  of  two  quantities 
/  and  p,  should  be  thought  of  as  a  single  physical  quantity,  just  as 
work,  the  product  of  F  and  s,  is  a  single  physical  quantity.  As 
such  we  shall  denote  it  by  L. 

83.  Work  Done  by  a  Moment  of  Force.  —  Let  the  line  of  action  of 
the  force  be  fixed  relatively  to  the  body,  so  that  the  moment  of 
force  about  the  axis  is  constant  and  equal  to 

fp.  If  the  body  rotate  through  an  angle  6 
in  the  direction  of  the  moment,  the  force 
will  have  acted  through  pd  and  the  work  done 
will  be  fpd  or  L0.  Hence  in  rotation, 

Work  =  moment  of  force  X  angular 

displacement  Fl0t  41- 

The  similarity  of  this  expression  to  that  for  the  work  done  in  trans- 
lation, namely  Fs,  should  be  noted,  moment  of  force  corre- 
sponding to  force  and  angular  displacement  to  linear  displacement. 

84.  Kinetic  Energy  of  Rotation.  —  Each  particle  of  a  rotating 
body  has   a  certain  linear  velocity  and   a  certain  amount  of 
kinetic  energy,  and  the  total  kinetic  energy  is  the  sum  of  the  ki- 
netic energies  of  the  particles.     A  particle  of  mass  m  at  a  dis- 
tance r  from  the  axis  has  a  linear  velocity  cor  where  w  is  the 
angular  velocity,  and  its  kinetic  energy  (in  absolute  units)   is 
Jwco2r2.     Now  r  is  different  for  different  particles  but  o>  is  the 
same  for  all. 

Hence 


The  term  in  brackets  evidently  depends  on  the  mass  and  form  of 
the  body  and  on  the  particular  axis  of  rotation  considered.  If 
we  denote  it  by  7, 

#  =  }/co2 

an  expression  similar  to  that  for  kinetic  energy  of  translation, 
namely 


64        MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

86.  Moments  of  Inertia.  —  The  expression  denoted  above  by  / 
is  called  the  moment  of  inertia  of  the  body  about  the  particular 
axis  of  rotation.  It  may  be  defined  as  the  sum  of  the  products 
of  the  particles  by  the  squares  of  their  respective  distances  from  the 
axis  of  rotation  ,  or  briefly, 

7=Smr2. 

The  value  of  this  sum,  in  the  case  of  a  body  of  regular  geomet- 
rical shape,  can  be  expressed  in  terms  of  its  mass  and  its  linear 
dimensions.  For  example,  all  parts  of  a  thin  hoop  of  mass  M 
and  radius  r  may  be  regarded  as  being  at  the  same  distance  from 
its  .geometrical  axis  and  its  moment  of  inertia  about  that  axis  is, 
therefore,  Mr2.  Two  other  important  cases  are  the  following: 
For  a  circular  disk  about  its  geometrical  axis  (Fig.  42). 


For  a  thin  red  about  a  transverse  axis  through  its  center  (Fig.  43). 


\     FlO.  42.  FIQ.    43. 

The  derivation  of  such  formulae  is  best  performed  by  means  of  the  Integral 
Calculus. 

If  p  be  the  mass  per  unit  area  of  the  disk,  and  if  it  be  supposed  divided  into 
hoops,  7  is  the  integral  from  0  to  r  of  2irrdrprz  which  is  frrp*  or  $Mrz. 

If  P  be  the  mass  of  unit  of  length  of  the  rod,  7  is  the  integral  from  — 1/2 
to  1/2  of  pdrr*  which  is  TV/8  or  &MI*. 

If  the  moment  of  inertia  of  a  body  about  a  certain  axis  is  7  and  the 
mass  of  the  body  is  M,  and  if  we  take  a  length  k  such  that  I=Mk*,  k  is 
called  the  radius  of  gyration  of  the  body  about  that  axis. 

86.  Energy  Equation  for  Rotation. — Consider  a  rigid  body  on  a 
fixed  axis  and  acted  on  by  a  moment  of  force  L,  the  moment  of 
inertia  about  the  axis  being  7.  If  the  body  turn  through  an  angle 
6  the  work  done  will  be  Z/0.  Since  the  body  is  rigid,  the  relative 
positions  of  its  particles  will  not  be  changed  and  there  will, 
therefore,  be  no  change  of  potential  energy.  Hence  the  work 
done  will  equal  the  increase  of  kinetic  energy,  or 


DYNAMICS  OF  RIGID  BODIES  65 

87.  Angular  Momentum.  —  In  the  case  just  considered  let  the 
time  of  rotation  through  the  angle  6  be  t.  The  average  angular 
velocity  in  the  time  t  is  }(w  +  coo).  Hence 


From  this  and  the  equation  of  §86  we  get 

Lt  =  !((#  —  WQ) 

The  product  Lt  evidently  corresponds  to  the  product  Ft  in  the 
case  of  translation  (§45)  and  may  be  called  the  impulse  of  the 
moment  of  force.  The  expression  on  the  right  is  the  increase  of 
/«.  From  the  analogy  of  momentum,  mv,  the  product  of  moment 
of  inertia  and  angular  velocity  is  called  angular  momentum. 

If  L  =  0,  that  is,  if  the  body  is  not  acted  on  by  any  force  having 
a  moment  about  the  axis  of  rotation,  7a>  =  /a>o,  or  the  angular 
momentum  is  constant. 

88.  Conservation  of  Angular  Momentum.  — 

The  last  statement  is  a  particular  case  of  the  principle  called  the  Conserva- 
tion of  Angular  Momentum,  namely,  the  case  of  a  rigid  body  mounted  on  a 
fixed  axis.  A  wobbling  quoit  is  a  more  general  case.  Neglecting  air  resist- 
ance, the  only  external  force  is  gravity,  which  acts  through  the  center  of 
mass.  The  angular  momentum  is  constant  in  amount  but  its  axis  has  a 
periodic  motion.  An  acrobat  turning  in  the  air  and  the  projectile  from  a 
rifled  gun  are  other  illustrations.  The  whole  solar  system  illustrates  the 
general  principle,  which  is  that  the  total  angular  momentum  of  any  system  of 
bodies  not  acted  on  by  forces  having  a  resultant  moment  about  the  center  of 
mass  is  constant  in  amount. 

89.  Angular  Acceleration  Produced  by  a  Moment  of  Force.  — 
From  §87  it  follows  that 


t 

Hence  if  a  be  the  angular  acceleration  produced  by  L 

L  =  Ia 

This  is  the  fundamental  equation  for  calculating  the  motion  of  a 
body  on  a  fixed  axis.     It  corresponds  to  F  =»  ma  for  translation. 

If  the  body  is  not  on  a  fixed  axis  but  is  free,  the  applied  force  will  produce 

linear  acceleration  of  the  center  of  mass  (§79)  and  also  rotation  about 

some  axis  through  the  center  of  mass,  the  direction  of  which  depends  on  the 

shape  of  the  body.    If  the  line  of  action  of  the  force  is  in  the  plane  of  two  of 

ft 


66          MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

the  principal  axes  (§101),  the  rotation  will  be  about  the  third,  but  the  sub- 
ject cannot  be  discussed  fully  here. 

90.  Moments  of  Inertia  about  Parallel  Axes. — There  is  a  simple 
and  useful  relation  between  the  moment  of  inertia,  7,  of  a  body 
about  any  axis  and  its  moment  of  inertia,  70,  about  a  parallel 
axis  through  the  center  of  mass,  namely, 


M  being  the  mass  of  the  body  and  h  the  dis- 
tance between  the  two  axes.  The  proof  of 
this  is  as  follows: 

Let  the  axes  referred  to  be  perpendicular  to 
FIG  44  *ke  plane  °f  Fig.  44  and  cut  the  plane  in  0 

and  C  respectively.  Let  the  coordinates  of 
a  particle,  m,  referred  to  rectangular  axes  through  C  be  3  and  y 
and  let  the  coordinates  of  C  referred  to  a  parallel  set  of  axes 
through  0  be  x  and  y.  Then 


70+ 

The  difference  between  the  right  hand  sides  is  zero,  since  x  and  y 
are  distances  from  planes  through  the  center  of  mass  and  (§76) 


91.  Kinetic  Energy  of  a  Body  which  has  Translation  and  Rota- 
tion. —  Let  the  body  of  Fig.  44  be  in  rotation  about  the  axis 
through  0,  supposed  fixed  in  the  body  (or  in  fixed  connection  with 
it)  with  angular  velocity  o>.  This  is  also  its  angular  velocity 
about  the  axis  through  C,  since  both  revolutions  are  completed 
in  the  same  time.  The  total  kinetic  energy  is  £7«2.  From  the 
equation  of  §90, 


since  F,  the  linear  velocity  of  C,  is  equal  to  hw.     Thus  the  total 
kinetic  energy  may  be  regarded  as  consisting  of 

(1)  K.  E.  of  Translation  of  the  C.  of  M. 

(2)  K.  E.  of  Rotation  about  the  C.  of  M. 

This  applies  to  a  cylinder  rolling  down  a  plane  or  a  moving  car- 
riage wheel,  since  there  is  an  axis  which  is,  for  the  moment,  at  rest, 


DYNAMICS  OF  RIGID  BODIES 


67 


namely,  the  line  of  contact  with  the  surface.  It  holds,  likewise, 
for  any  body  rotating  about  an  axis  through  the  center  of  mass 
and  also  moving  perpendicular  to  that  axis,  for  there  is  in  all 
such  cases  an  instantaneous  axis  (§69)  .  In  fact,  by  resolving  the 
linear  velocity  parallel  to  and  perpendicular 
to  the  axis,  it  will  be  seen  that  the  principle 
is  true  for  any  motion  of  a  rigid  body. 

92.  Moments  of  Inertia  of  a  Disk.—  If  the 
moment  of  inertia  of  a  disk  of  any  shape  about 
two  axes  at  right  angles  in  the  plane  of  the 
disk  be  /i  and  72,  its  moment  of  inertia  about 
a  third  axis  intersecting  the  two  and  perpendicular  to  the  plane 
of  the  disk  is  /i  +  /2. 

For  let  the  distances  of  an  element,  m,  of  the  disk  from  the 
first  two  axes  be  TI  and  r2  respectively  Fig.  45. 


mr 


Summing  up  for  all  elements 


Fio.  45. 


93.  Some  other  Moments  of  Inertia.  — 

A  uniform  rectangular  disk  may  be  divided  into  rods  and  their  moments 

of  inertia  added.     Thus  about  axes  in  the  plane  of  the  disk  and  bisecting 

pairs  of  opposite  sides  I9=^Ma2,  Ib=^Mb*.     Hence    the    moment    of 

inertia  of  the  disk  about  an  axis  through  the  center 

A  perpendicular  to  the  disk  is 


A  uniform  rectangular  block  may  be  divided 
into  rectangular  disks.  Hence  its  moment  of  in- 
ertia about  an  axis  through  its  center  and  perpen- 
dicular to  a  face  with  sides  equal  to  a  and  b  is  given 
by  the  last  formula. 

The  moment  of  inertia  of  a  uniform  circular  disk 

about  any  two  axes  at  right  angles  in  the  plane  of  the  disk  are  equal. 
Hence  by  §92,  each  is  equal  to  JAfr*. 

A  circular  cylinder  may  be  divided  into  circular  disks.  Hence  its  moment 
of  inertia  about  its  geometrical  axis  is  £M  r*.  Its  moment  of  inertia  about  a 
transverse  axis  through  its  center  may  also  be  found  by  the  above  principles, 
but  we  leave  it  as  an  exercise. 


68 


MECHANICS  AND  THE  PROPERTIES  OF  MATTER 


94.  Gravitational  Units. — All  of  the  preceding  formulae  are  in 
absolute  units.  To  adapt  them  to  gravitational  units,  it  is  only 
necessary  to  replace  m  and  M,  wherever  they  occur  by  m/g  and 
M/g.  For  at  the  outset  (§84)  we  took  Jrav2  as  the  kinetic 


m 


energy  of  m,  whereas  in  gravitational  units  it  would  be  J  — 

(Engineers  write  W  instead  of  m).     All  formulae  in  which  mass 
does  not  occur  explicitly  are  the  same  in  both  systems. 


TABLE  or  MOMENTS  OP  INERTIA 


Body 


Axis 


Moment  of  Inertia 


Rod 
Rod 

Circular  disk 
Circular  cylinder 
Circular  cylinder 
Rectangular  block 

Sphere 


transverse  through  end 
transverse  through  middle 
perpendicular  through  center 
longitudinal  through  center 
transverse  through  center 
through  center  perpendicular  to 
face  with  sides  a  and  b  in  length 
through  center 


RESULTANT  OF  FORCES  ACTING  ON  A  BODY 

95.  Resultant. — When   treating   of  the   forces   acting  on   a 
particle,  we  found  that  they  could  always  be  replaced  by  a  single 
equivalent  force  called  their  resultant.     When  a  number  of  forces 
act  on  a  body,  they  are  in  certain  cases  equivalent  in  their  effects 
to  a  single  force,  which  is  called  their  resultant.     As  we  shall  see 
later,  there  are  other  cases  in  which  this  is  not  so. 

96.  Conditions  to  be  Satisfied  by  Resultant.— 1.  The  resultant 
must  be  competent  to  produce  the  actual  linear  acceleration  of 
the  center  of  mass  C,  and,  therefore,  its  component  in  any  direc- 
tion must  equal  the  sum  of  the  components  of  the  acting  forces  in 
that  direction.     This  condition  is  simplified  by  considering  that 
any  actual  acceleration  of  C  is  made  up  of  three  independent  com- 
ponents along  axes  at  right  angles.     Hence  the  resultant  must 
have  a  component  in  each  of  three  rectangular  directions  equal  to  the 
sum  of  the  components  of  the  forces  in  these  directions. 

2.  The  resultant  must  be  competent  to  produce  the  actual  an- 


DYNAMICS  OF  RIGID  BODIES 


69 


We 


gular  acceleration  about  any  axis,  and,  therefore,  its  moment 
about  any  axis  must  equal  the  sum  of  the  moments  of  the  acting 
forces  about  that  axis.  It  is,  however,  not  necessary  to  consider 
all  axes;  for  the  angular  acceleration  about  any  axis  can  be 
resolved  into  rectangular  components.  Hence  the  second  con- 
dition is  that  the  moment  of  the  resultant  about  each  of  any  set 
of  rectangular  axes  must  equal  the  sum  of  the  moments  of  the  forces 
about  that  axis. 

If  a  force  satisfies  the  above  conditions  it  is  the  resultant. 
shall  now  apply  these  tests  to  find  the  re- 
sultant of  the  forces  acting  on  a  body  in  some 
cases  of  importance. 

97.  Resultant  of  Two  Parallel  Forces.—  1. 
Let  P  and  Q  be  two  forces  in  the  same  direc- 
tion acting  at  points  A  and  B  of  a  body. 
A  single  force  R  in  the  direction  of  P  and  Q 
and  equal  to  (P  +  Q)  will  satisfy  the  first  con- 
dition of  §96,  since  its  component  in  any 
direction  equals  the  sum  of  the  components  of 
P  and  Q  in  that  direction. 

This  force  will  also  satisfy  the  second  condition,  provided  it 
acts  at  a  point  C  in  AB  such  that 

P.CA=Q-CB 

For  first  consider  any  axis  perpendicular  to  the  plane  of  P  and 
Q.  Suppose  it  to  cut  that  plane  in  0.  Draw  OA'C'B'  to  cut 
the  lines  of  the  forces  at  right  angles.  Since  C'  A.'  and  C'B'  are 
the  projections  of  CA  and  CB  respectively  it  is  readily  seen  from 
the  last  equation  that 


or 


Thus  the  moment  of  R  about  the  axis  equals  the  sum  of  the 
moments  of  P  and  Q.  Next  take  an  axis  perpendicular  to  the 
above  axis  and  to  the  lines  of  the  forces.  All  the  forces  are  at  the 
same  distance  from  this  axis,  and,  since  R  equals  (P  +  Q),  the 
moment  of  R  about  it  equals  the  sum  of  the  moments  of  P  and  Q 
about  it.  Finally  an  axis  perpendicular  to  the  other  two  will  be 
parallel  to  P,  Q,  and  R  and  each  will  have  zero  moment  about  it. 


P(OC"  -  OA')  =  Q(OB'  -  0C') 


70          MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

Hence  R  is  the  resultant  of  P  and  Q. 

It  is  important  to  notice  that  C  is  the  point  we  would  have 
found  if  we  had  been  seeking  the  center  of 
mass  of  particles  at  A  and  B  proportional 
to  P  and  Q  (§74). 

2.  Let  P  and  Q  be  in  opposite  directions 
and  suppose  P>Q.    A  single  force  R  in  the 
„  direction  of  the  greater,  P,  and  equal  to 

C  (P  —  Q)  will  satisfy  the  first  condition,  since 

FIO.  48.  its  component  in  any  direction  equals  the 

(algebraic)  sum  of  the  components  of  P 
and  Q  in  that  direction.  It  will  also  satisfy  the  second  condition 
if  it  acts  at  a  point  C  in  BA  produced  such  that 


C 


For  consider  first  an  axis  perpendicular  to  the  plane  of  the  forces 
and  cutting  that  plane  in  0,  and  draw  OC'A'B'  to  cut  the  lines 
of  the  forces  at  right  angles.  Then 


or  P(OA'  -  0C")  =Q(OB'  -  OC') 


Hence  the  moment  of  R  about  the  axis  equals  the  (algebraic) 
sum  of  the  moments  of  P  and  Q.  The  same  is  true  of  two  other 
axes  taken  as  in  case  (1) ;  the  proof  need  not  be  repeated  as  it  is 
identical  with  that  there  given. 

Hence  R  is  the  resultant  of  P  and  Q. 

To  find  the  distance  of  C  from  A  replace  CB  by  (CA+AB) 
in  the  above  equation  for  the  position  of  C.  Then 

_Q-AB 
'P-Q 

Hence  CA  is  greater  the  less  (P  — Q),  that  is,  the  more  nearly 
the  forces  are  equal. 

98.  Couples, — Two  equal  and  opposite  forces,  not  in  the  same 
line,  constitute  a  couple.  If  we  attempted  to  find  the  resultant  of 
two  such  forces  by  the  method  of  the  last  section,  it  would 
give  zero  force  at  an  infinite  distance  and  such  a  force  has  no  real 
existence.  Hence  a  couple  cannot  be  reduced  to  a  single  force. 


DYNAMICS  OF  RIGID  BODIES  71 

The  sum  of  the  moments  of  two  forces  constituting  a  couple 
is  the  same  about  all  axes  perpendicular  to  the  plane  of  the 
couple.  For,  about  an  axis  0  between  the  forces,  the  moments 
of  the  forces  are  in  the  same  direction  and  their  sum  is 
(P'OA+P-OB)  or  P-AB;  and,  about  an  axis  0' 
not  between  the  forces,  the  moments  are  in  op-  , 


~ 


posite  directions  and  the  sum  is  (P-O'B'  -  P-O'A')  -  -  -  B( 

which  again  equals  P-AB.  A 

The  distance  A  B  between  the  forces  of  a  couple        y 
is  sometimes  called  the  arm  of  a  couple,  and  the        Fia  49< 
moment  of  the  couple  about  any  axis  perpendicular 
to  its  plane,  that  is  P-ABt  is  sometimes  called  the  strength  of 
the  couple.     Two  couples  in  the  same  or  parallel  planes  and 
of  the  same  strength  are  equal  in  all  respects  and  produce 
equal  effects. 

Since  the  sum  of  the  forces  of  a  couple  equals  zero,  the  couple 
produces  no  acceleration  of  the  center  of  mass  (§79)  ;  and  if  the 
center  of  mass  be  at  rest  it  will  remain  at  rest,  or  if  it  be  moving 
in  any  way  it  will  continue  moving  with  constant  velocity.  The 
angular  velocity  produced  by  the  couple  must  therefore  be  about 
some  axis  through  the  center  of  mass. 

99.  Resultant  of  any  Number  of  Parallel  Forces.  —  To  find  the 
resultant  of  any  number  of  parallel  forces,  whether  in  one  plane 
or  not,  we  may  find  the  resultant  of  two,  then  combine  this 
resultant  with  a  third,  and  so  on.     The  final  resultant  will  be 
either  a  single  force  or  a  couple  or  zero.     At  each  step  the  result- 
ant equals  the  algebraic  sum  of  the  forces  added.     Hence  the 
final  resultant  equals  the  algebraic  sum  of  all  the  forces. 

The  line  of  action  of  the  resultant  may  also  be  found  by  apply- 
ing the  principle  that  the  moment  of  the  resultant  about  any  axis 
must  equal  the  sum  of  the  moments  of  the  forces  about  that  axis. 
When  the  forces  are  all  in  one  plane,  to  find  the  line  of  action  of 
the  resultant  we  only  need  to  take  moments  about  any  axis  per- 
pendicular to  the  plane.  When  the  forces  are  not  all  in  one  plane, 
it  will  be  necessary  to  take  moments  about  two  rectangular  axes 
perpendicular  to  the  forces. 

100.  Center  of  Gravity.  —  Attention  has  been  called  in  (1)  §97 
to  the  identity  of  the  method  of  finding  the  resultant  of  parallel 
forces  in  the  same  direction  and  the  method  of  finding  the  center 


72          MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

of  mass  of  a  number  of  particles.  If,  for  the  particles  in  a  certain 
group  of  particles  or  of  a  body,  we  substitute  parallel  forces  all 
in  one  direction  acting  at  the  respective  positions  of  the  particles 
and  proportional  to  the  masses  of  the  particles,  the  point  of  action 
of  the  resultant  will  coincide  with  the  center  of  mass.  This  is 
sometimes  taken  as  the  definition  of  the  center  of  mass.  It 
should  be  noticed  that  nothing  need  be  said  as  to  the  common 
direction  of  the  parallel  forces. 

The  forces  of  gravity  on  the  particles  of  a  body  are  (very 
nearly)  parallel  forces  and  they  are  proportional  to  the  masses  of 
the  particles.  Hence  the  Center  of  Gravity  of  a  body,  or  the 
point  of  action  of  the  resultant  of  the  (very  nearly)  parallel 
forces  of  gravity,  coincides  with  the  center  of  mass  of  the  body. 

A  very  large  body  near  the  earth  has  a  definite  center  of  mass  but  not  a 
definite  center  of  gravity  (except  in  some  particular  cases),  for  the  forces 
are  not  quite  parallel  nor  quite  proportional  to  the  masses.  This  is  of  no 
practical  importance  as  regards  bodies  of  the  size  found  on  the  earth's 
surface;  but  it  is  of  great  importance  in  considering  the  effect  of  the  attrac- 
tion of  the  sun  and  moon  on  the  motion  of  the  earth. 

101.  Centrifugal  Force. — In  §47  we  found  an  expression  for 
the  force  required  to  keep  a  particle  revolving  in  a  circle.  We 
may  now  extend  this  to  a  body  of  any  size  or  shape.  When  a 
body  of  mass  m  rotates  with  constant  angular  velocity  about  any 
axis  not  through  the  center  of  mass,  the  latter  moves  uniformly 
in  a  circle  and  has  therefore  an  acceleration  v*/r  toward  the  cen- 
ter. Hence  the  force  acting  on  the  body  (or  the  resultant  of  the 
forces  if  there  are  several)  must,  by  the  principle  stated  in  §79, 
equal  mv*/r  and  must  act  in  the  line  joining  the  center  of  mass 
to  the  center  of  the  circle,  and  the  body  will  react  with  an  equal 
and  opposite  force.  This  reaction  is  the  cause  of  the  varying 
force  which  an  unbalanced  fly-wheel  exerts  on  the  axis. 

In  many  cases  more  than  a  single  force  (in  addition  to  those  required  to 
overcome  friction  and  sustain  the  weight  of  the  body)  is  required  to  keep 
a  body  rotating  about  an  axis.  As  a  simple  case  consider  a  pair  of  equal 
spheres  joined  by  a  light  rod  and  rotating  about  a  vertical  axis  through  the 
center  of  the  rod.  Since  the  center  of  mass  has  no  acceleration,  the  forces 
acting  on  the  body  if  transferred  to  the  center  of  mass  would  have  a  zero 
resultant.  Hence  the  forces  must  form  a  couple  and  the  reactions  on  the 
axis  will  form  a  couple,  called  a  centrifugal  couple,  tending  to  bend  the  axis 
or  make  it  rotate  about  an  axis  perpendicular  to  itself.  For  certain  axes  of 


DYNAMICS  OF  RIGID  BODIES  73 

rotation  of  a  body  the  centrifugal  couple  is  zero.     In  the  above  simple 
illustration  this  is  true  when  the  axis  of  rotation  is  in  the  line  of  the  centers 
of  the  balls  or  at  right  angles  thereto.     These  are  also  the 
positions  of  maximum  and  minimum  moments  of  inertia  of  p 

the  body.     A  similar  statement  will  evidently  apply  to  a 
symmetrical  body,  such  as  a  circular  disk,  which  can  be 
divided  into  pairs  of  particles  like  the  above.     Whatever  the 
shape  of  a  body  there  are  three  rectangular  axes  through 
any  point  of  the  body  about  which  it  can  rotate  without  ex- 
erting any  centrifugal  couple.     These  are  the  axis  of  maxi-        Fio.  60. 
mum  moment  of  inertia  through  the  point,  that  of  minimum 
moment  of  inertia  and  'a  third  perpendicular  to  both.     These  are  called  the 
principal  axes  through  the  point. 

When  a  body  is  set  spinning  about  a  principal  axis  through  its  center 
of  mass  it  continues  to  spin  without  any  tendency  to  "wobble"  or  exert 
a  centrifugal  couple.  This  is  illustrated  by  the  motion  of  a  well-known 
quoit  or  discus,  by  that  of  a  bullet  from  a  rifled  gun  and  by  the  motion  of 
the  earth  about  its  axis.  But  when  the  axis  of  initial  spin  is  not  a  principal 
axis  irregular  motion  ensues,  as  is  illustrated  by  a  badly  thrown  quoit. 

FORCES  IN  EQUILIBRIUM 

102.  Conditions  of  Equilibrium. — The  forces  acting  on  a  body 
are  in  equilibrium  when  they  cause  no  acceleration  either  linear 
or  angular,  that  is  when  their  resultant  is  zero. 

Given  that  a  system  of  forces  is  in  equilibrium  we  may  con- 
clude (from  §79)  that  the  sum  of  their  components  in  any  direc- 
tion equals  zero,  since  there  is  no  acceleration  of  the  center  of 
mass,  and  also  that  the  sum  of  their  moments  about  any  axis 
equals  zero,  since  there  is  no  angular  acceleration  about  any  axis. 

When  we  equate  the  sum  of  the  components  of  the  forces  in  any 
direction  to  zero  we  get  a  relation  between  the  forces,  and  it 
might  seem  that  we  could  get  an  unlimited  number  of  such  rela- 
tions; but,  in  reality,  there  are  only  three  of  these  independent, 
e.g.,  those  got  by  taking  the  sum  of  the  components  in  some  three 
directions  at  right  angles. 

Similarly  we  get  a  relation  between  the  forces  by  equating  the 
sum  of  the  moments  about  any  axis  to  zero;  but  again  there  are 
only  three  of  these  relations  independent,  e.g.,  those  got  by  taking 
moments  about  some  three  rectangular  axes. 

Thus  we  can  deduce  at  most  six  independent  relations  between 
forces  in  equilibrium,  and  this  might  have  been  expected  from  the 
fact  that  a  rigid  body  has  six  degrees  of  freedom  at  most — three 
of  translation  and  three  of  rotation. 


74          MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

We  may  reverse  the  point  of  view  and  ask  what  relations  and 
how  many  must  forces  satisfy  to  make  it  certain  that  they  shall  be 
in  equilibrium,  that  is,  what  are  the  conditions  essential  to  equi- 
librium. The  answer  is  again  six  relations,  namely,  the  sum  of 
the  components  in  each  of  any  three  rectangular  directions  must 
equal  zero  and  the  sum  of  the  moments  about  each  of  some  three 
rectangular  axes  must  equal  zero. 

The  two  conditions  may  be  stated  thus: 


for  each  of  any  three  rectangular  axes. 

103.  Forces  in  a  Plane.  —  When  the  lines  of  action  of  forces  that 
are  in  equilibrium  lie  in  one  plane,  the  sum  of  the  components  of 
the  forces  in  each  of  any  two  directions  at  right  angles  in  the 
plane  equals  zero.  In  this  case  the  third  rectangular  axis  is  per- 
pendicular to  the  plane  and  the  component  of  each  force  in  that 
direction  is  zero.  Also  the  sum  of  the  moments  of  the  forces 
about  any  axis  perpendicular  to  the  plane  is  zero.  The  other 
two  rectangular  axes  are  in  the  plane  and  the  moment  of  any 

one  of  the  forces  about  such  an  axis  is  zero. 
Hence  when  forces  in  a  plane  are  in  equilib- 

rium   three    independent    relations    among   the 

forces  can  be  deduced. 


104.   Examples  of  Equilibrium  of  Forces  in  a  Plane. — 
To  illustrate  the  above  we  shall  consider  two  examples. 

1.  A  uniform  beam  AB  (length -»Z)  rests  without  slip- 
ping on  the  ground  and  leans  without  friction  against  a 
smooth  wall.  What  is  the  force  (FJ  at  the  wall  and  the 
vertical  force  at  the  ground  (F3)  and  what  is  the  force  of  friction  (Ft)  be- 
tween the  beam  and  the  ground  (Fig.  51)? 

Since  there  is  no  friction  at  B,  Fl  is  horizontal.  The  force  of  friction 
at  A,  that  is  Fs,  is  horizontal  and  toward  J£.  Equating  the  sum  of  the 
horizontal  forces  acting  on  the  beam  to  zero  we  get 

Fi-Ft-Q.  (1) 

and  equating  the  vertical  forces  to  zero  we  get 

J^-TF-0  (2) 

A  third  relation  may  be  obtained  by  taking  moments  about  any  axis  perpen- 
dicular to  the  plane  of  the  forces.  If  we  choose  for  this  purpose  an  axis 
through  A,  the  relation  will  be  as  simple  as  possible,  since  F^  and  JP,  have 


DYNAMICS  OF  RIGID  BODIES  75 

zero  moment  about  such  an  axis.  The  weight  acts  at  the  center  C  of 
the  beam  and  the  distance  of  its  line  of  action  from  A  is  (1/2)  cos  0.  Also 
the  distance  BE  of  the  line  of  action  of  Fl  from  A  equals  I  sin  6.  Hence 

W  —  cos  0-FJ  sin  6  =  0  , 

2  (o) 

From  these  three  equations  we  get 

Fi—Ft  —  W  cot  6 

2.  A  uniform  rod  hangs  from  a  wall  by  a  hinge  and  rests  on  a  smooth 
floor  (Fig.  52).     In  this  case  the  force  at  A  must  be  vertical,  since  there  is 
no  horizontal  force  of  friction  at  A.    Let  the  force  on  the  beam  at  B  consist 
of  a  horizontal  part  Fl  and  a  vertical  part  Ft.    Equat- 
ing to  zero  the  sum  of  the  vertical  forces,  the  sum  of      £» 
the  horizontal  forces  and  the  sum  of  the  moments  about 
B,  we  get 


W  —cos  d-FJ  cos  0=0 


Hence 


Since  Fl  is  zero  the  rod  does  not  press  against  the 
wall.     This  result,  which  seems  at  first  improbable,  may 
be  verified  by  allowing  A  to  rest  on  a  board  on  a  tank  of  water  and  hang- 
ing B  by  a  cord;  the  cord  will  be  found  to  be  vertical  when  tested  by  com- 
parison with  a  plumb  line. 

105.  Special  Cases  of  Equilibrium. — 1.  When  two  forces  are  in 
equilibrium  they  must  be  equal  and  opposite  and  in  the  same  line. 
If  not  equal  and  opposite  they  would  produce  translation,  and  if 
not  in  the  same  line  they  would  produce  rotation. 

For  example,  a  body  suspended  by  a  cord  must  rest  so  that  its 
center  of  gravity  is  vertically  below  the  point  of  support.  This 
supplies  an  experimental  method  of  finding  the  center  of  gravity 
of  a  disk  of  any  shape.  It  is  only  necessary  to  support  it  in 
succession  at  two  points  on  its  rim  and  find  the  intersection  of  the 
lines  of  support. 

2.  When  three  forces  are  in  equilibrium  they  must  all  lie  in  one 
plane.  For  the  sum  of  the  moments  of  all  three  about  any  axis  is 
zero.  About  any  axis  that  intersects  the  lines  of  action  of  two 
of  the  forces  the  moments  of  these  two  forces  are  zero.  Hence 
any  such  axis  must  also  intersect  the  line  of  action  of  the  third 
force  (unless  it  be  parallel  to  it)  and  this  cannot  be  so  unless  all 
the  forces  lie  in  one  plane. 


76          MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

3.  Three  forces  in  equilibrium  must  either  be  parallel  or  pass 
through  a  single  point.  If  they  are  parallel,  one  is  equal  and 
opposite  to  the  resultant  of  the  other  two.  If  they  are  not 
parallel,  two  of  them  intersect  and  their  moments  about  the 
point  of  intersection  are  zero.  Hence  the  third  must  pass 
through  the  point  of  intersection  of  any  two. 

As  an  example  of  three  parallel  forces  in  equilibrium  consider  (2)  of  §104. 
The  resultant  of  Fa  and  F3  must  be  equal  and  opposite  to  and  in  same 
line  as  W  which  acts  at  the  middle  of  AB.  Hence  Ft  and  Ft  are  equal. 

As  an  example  of  three  non-parallel  forces  in  equilibrium  consider  (1) 
of  §104.  Let  the  resultant  of  Fa  and  Ft  be  F.  Then  F,  Fl  and  W  are 
three  forces  in  equilibrium.  Hence  F  must  pass  through  the  intersection 
of  Fl  and  W.  Hence  the  direction  of  F  is  readily  found  graphically.  We 
may  also  find  graphically  the  magnitudes  of  Fl  and  F.  Since  DA  and  BH 
are  equal,  HBDA  is  a  parallelogram.  Hence  F,  F^  and  W  are  proportional 
to  HA,  HB  and  HD. 

106.  Stable,  Unstable  and  Neutral  Equilibrium. — A  body  is  in 
equilibrium  when  it  is  either  at  rest  or  moving  uniformly,  that  is, 


Fia.  53. — Stable,  unstable  and  neutral  equilibrium. 

without  acceleration  linear  or  angular.     The  resultant   of   the 
forces  acting  on  such  a  body  is  zero. 

When  a  body  in  equilibrium  is  at  rest  the  equilibrium  is  de- 
scribed as  static.  Of  this  kind  of  equilibrium  there  are  three 
forms,  stable,  unstable  and  neutral.  A  body  at  rest  is  in  stable 
equilibrium  when,  on  being  slightly  displaced,  it  tends  to  return 
to  its  equilibrium  position.  This  is  illustrated  by  a  chemical 
balance,  a  pendulum  or  picture  hanging  by  a  cord,  a  book  on  a 
table  and  in  fact  by  most  stationary  objects.  A  body  at  rest  is 
in  unstable  equilibrium  when,  on  being  slightly  displaced,  it  tends 
to  move  further  from  its  equilibrium  position.  An  egg  on  end 
and  a  board  balanced  on  one  corner  would  be  in  unstable  equi- 
librium. A  body  at  rest  is  in  neutral  equilibrium  when,  on  being 
slightly  displaced,  it  has  no  tendency  either  to  move  further  away  or 


DYNAMICS  OF  RIGID  BODIES  77 

to  return;  for  example,  a  sphere  or  cylinder  on  a  horizontal  table 
and  any  body  mounted  on  an  axis  through  its  center  of  gravity. 

A  body  in  a  position  of  stable  equilibrium  oscillates  about  that 
position  when  displaced  and  released,  though  the  oscillation  may 
be  quickly  destroyed  by  friction  or  other  forces.  When  too  far 
displaced  such  a  body  may  come  to  a  position  of  unstable  equi- 
librium and  not  return;  a  table  or  chair  tilted  too  far  comes  to  a 
position  of  unstable  equilibrium.  The  extent  to  which  any  such 
body  may  be  displaced  and  yet  return  is  a  measure  of  the  degree 
of  stability  of  the  equilibrium. 

107.  Energy  Test  of  Static  Equilibrium. — When  a  body  at  rest 
is  in  stable  equilibrium,  a  disturbance  will  increase  its  potential 
energy.  This  is  evident  in  the  case  of  a  pendulum  'at  rest,  for  a 
disturbance  raises  its  center  of  gravity;  work  is  done  against 
gravity  when  the  body  is  displaced  and  this  work  produces  poten- 
tial energy.  Thus  a  position  of  stable  equilibrium  is  a  position  in 
which  the  potential  energy  is  a  minimum.  This  statement  holds 
true  whatever  the  force  against  which  work  is  done;  the  fact  that 
the  disturbance  produces  an  increase  of  potential  energy  shows 
that  there  are  conservative  forces  opposing  the  motion  and  these 
forces  will  cause  the  body  to  return  when  it  is  displaced. 

A  position  of  unstable  equilibrium  is  a  position  in  which  the 
potential  energy  is  a  maximum ,  as  is  illustrated  by  a  spheroid  on 
end  or  a  board  balanced  on  a  corner;  a  disturbance  lowers  the 
center  of  gravity.  The  statement  is  true  whatever  the  forces  in 
action;  for  the  fact  that  the  body  when  disturbed  moves  farther 
away  from  its  position  of  equilibrium  and  thus  gains  kinetic 
energy  shows  that  its  potential  energy  diminishes. 

When  the  equilibrium  is  neutral  a  displacement  produces  no 
change  of  potential  energy;  when  a  sphere  rolls  on  a  horizontal 
table  its  center  neither  rises  nor  falls.      An 
interesting  illustration  is  afforded  by  the  ap- 
paratus sketched  in  the  adjoining  figure.     It 
will  remain  at  rest  whatever  the  positions  of 
the  equal  weights  which  are  adjustable  along 
the  horizontal  rods,  for  the  total  potential 
energy  fa  the  same  in  all. 

A  body,  such  as  a  fly  wheel  or  a  railway 
car,  in  a  steady  state  of  motion  is  in  kinetic  equilibrium  since 
the  resultant  of  the  forces  acting  on  it  is  zero. 


78          MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

The  principle  that  for  stable  equilibrium  the  potential  energy  is  a  mini- 
mum is  extensively  illustrated  in  nature;  the  potential  energy  may  be 
partly  or  wholly  other  than  mechanical  energy,  in  forms  dealt  with  in 
other  parts  of  Physics.  Changes  are  continually  taking  place  in  nature 
and  bodies,  when  disturbed,  settle  into  states  of  stable  equilibrium,  that 
is,  of  minimum  potential  energy. 

KINEMATICS  AND  DYNAMICS 
PERIODIC  MOTIONS 

108.  A  periodic  motion  is  one  that  is  repeated  in  successive 
equal  intervals  of  time.     The  time  required  for  each  such  repeti- 
tion is  called  the  period  of  the  motion.     Thus,  the  moon  revolves 
around  the  earth  with  a  periodic  motion,  the  period  of  which  is  a 
lunar  month  and  the  earth  revolves  about  the  sun  in  a, period  of  a 
year.     The  end  of  a  hand  of  a  clock  has  a  periodic  motion  about 
the  center  of  the  face.     A  point  on  a  vibrating  violin  string  or 
piano  wire  has  a  periodic  motion. 

109.  Uniform  Circular  Motion. — When  a  point  P  revolves  with 
constant  speed  in  a  circle  of  center  0,  the  position  of  P  at  any 
moment  may  be  assigned  by  giving  the  angle  that  OP  makes 

with  some  fixed  diameter  such  as  OA.     This 
angle  is  called  the  phase  of  P's  motion. 

If  the  period  of  the  motion  is  T,  the  angle 
'through  which  OP  revolves  in  unit  time  is  the 
I    angular  velocity  a)  and  equals  2n/T.     Let  us 
suppose  that  at  the  moment  from  which  we 
begin  reckoning  time  P  is  at  some  position  B, 
FIQ  55  and  let  its  phase  at  that  moment,  that  is,  the 

angle  BOA,  be  e.  After  time  t,  P  will  have 
revolved  through  an  angle  cat  or  (2n/T)t  and  the  phase  at  time 
t  will  be  [(2n/T)t  +  e]. 

110.  Simple  Harmonic  Motion. — This  is  the  most  important 
form  of  periodic  motion  and  is  illustrated  by  the  vibration  of  a 
simple  pendulum  swinging  in  a  small  arc,  of  a  weight  hung  from  an 
elastic  cord  or  spring  and  moving  vertically,  of  a  point  on  the 
prong  of  a  tuning  fork  and  many  other  cases. 

Simple  harmonic  motion  is  a  linear  vibration,  the  motion  being 
such  that  the  vibrating  point  has  an  acceleration  which  is  toward  the 
center  of  its  path  qnd  proportional  to  its  distance  from  the  center. 


KINEMATICS  AND  DYNAMICS  79 

Let  A!  A  be  the  path  of  vibration  of  a  point  M  which  has  a 
simple  harmonic  motion,  and  let  C  be  the  center  of  A'  A.  Denote 
the  distance  of  M  from  C  at  any  time  by  x,  and  let  values  of  x  be 
considered  as  positive  when  M  lies  between  C  and  A  and  negative 
when  M  lies  between  C  and  A'.  When  a;  is  c  M 

positive  the  acceleration,  a,  of  M  is  toward      j;  '    x    ~~A 

C  and  is,  therefore,  in  the  negative  direc-  Fio  5Q 

tion,  and  when  x  is  negative  a,  being  still 
toward  C,  is  positive.     Hence,  if  we  denote  the  constant  of  pro- 
portionality of  the  magnitude  of  a  to  x  by  c,  by  the  above  defi- 
nition of  simple  harmonic  motion 

a=  —  ex 

The  distance  x  of  the  vibrating  point  from  the  center  of  motion  is 
called  the  displacement  of  the  point. 

One-half  of  the  length  of  the  path  of  vibration  is  catted  the  ampli- 
tude of  the  simple  harmonic  motion.  We  shall  denote  it  by  r. 
It  is  equal  to  the  magnitude  of  the  greatest  displacement  (CA 
or  CA'). 

The  time  required  for  a  complete  vibration  (that  is,  from  A  to  A' 
and  back  to  A)  is  the  period  of  the  simple  harmonic  motion. 

111.  The  Force  Acting  on  a  Body  that  has  Simple  Harmonic 
Motion.  —  A  body  that  has  a  simple  harmonic  motion  has  a  vary- 
ing acceleration  which  is  always  directed  toward  the  center.  To 
produce  this  acceleration  a  varying  force,  also  directed  toward 
the  center,  must  act  on  the  body.  Denote  the  force  by  F  and  let 
m  be  the  mass  of  the  body.  From  the  Second  Law  of  Motion  and 
the  definition  of  simple  harmonic  motion  we  get 


=  —mcx 

Since  m  and  c  are  constants  for  a  given  body  and  a  given  simple 
harmonic  motion,  the  force  required  is  always  opposite  to  and 
proportional  to  the  displacement. 

The  force  required  to  stretch  or  compress  a  spiral  spring,  one 
end  of  which  is  fixed,  is  proportional  to  the  displacement  of  the 
free  end  from  its  unstrained  position,  and  the  reaction  exerted  by 
the  spring  is  opposite  to  and  proportional  to  the  displacement 
(§56).  Hence  a  body  attached  to  such  a  spring  and  allowed  to 


80 


MECHANICS  AND  THE  PROPERTIES  OF  MATTER 


vibrate  under  the  action  of  the  spring  has  simple  harmonic 
motion.  The  same  law  of  force  holds  for  a  flat  spring  when  bent 
and,  in  fact,  for  any  elastic  body  when  distorted.  Hence  all 
elastic  vibrations  are  simple  harmonic  motions  or  compounded 
of  such  motions  and  the  same  is  true  of  the  vibrations  that  con- 
stitute sound  and  light. 

112.  Energy  of  a  Body  that  has  Simple  Harmonic  Motion.  —  The 
principle  of  the  Conservation  of  Energy  applies  to  a  body  that 
has  simple  harmonic  motion,  since  the  only  force  acting  on  the 
body  is  one  that  depends  on  the  position  of  the  body  (§63).     We 
have,  in  fact,  already  found  in  §61  the  proper  expression  for  the 
total  energy  of  such  a  body  in  any  position;  all  we  need  to  do  is  to 
substitute  for  k  its  value  in  the  present  case,  namely,  me.     Hence 
the  total  energy  is  ($mv2  +  $mcx2),  of  which  the  first  part  is  the 
kinetic  energy  at  displacement  x  and  the  second  is  the  potential 
energy.     At  one  end  of  the  path  of  vibration  v  is  zero  and  x  =  r; 
hence  the  total  energy  is  potential  and  equal  to  Jmcr3.     At  the 
center  x  is  zero  and  v  has  its  largest  value,  7;  hence  the  energy  is 
entirely  kinetic  and  equal  to  JmV2. 

113.  Velocity  in  Simple  Harmonic  Motion.  —  From  the  result 
just  stated  we  can  find  a  useful  expression  for  the  velocity  at  any 
displacement.     Equating  the  total  energy  at  displacement  x  to 
that  at  maximum  displacement  we  have 


and,  referring  to  Fig.  56,  it  will  be  seen  that  the  positive  sign  must 
be  taken  for  motion  from  A'  to  A  and  the 
negative  for  motion  from  A  to  A'. 

114.  Relation  Between  Simple  Harmonic 
and    Circular  Motions.  —  Simple   harmonic 
motion  has  been  defined  as  a  vibration  in  a 
line  according  to  the  law  o=»—  ex.     Now 
the  projection  of  a  uniform  circular  motion 
on  a  diameter  of  the  circle  has  exactly  the 
same  character.     For  on  A'  A  as  diameter 
draw  a  circle,  and  let  P  revolve  with  con- 
stant angular  velocity,  w,  in  the  circle.     If  M  is  the  projection  of 
P  on  A'  A,  M  vibrates  once  along  A'  A  in  each  revolution  of  P. 


KINEMATICS  AND  DYNAMICS  81 

Since  the  motion  of  M  is  that  part  of  the  motion  of  P  which  is  in 
the  direction  of  A' A,  the  acceleration,  a,  of  M  is  the  component 
of  the  acceleration  of  P  in  that  direction.  The  accceleration  of 
P  is  w2r  in  the  direction  PC  or  —  o>2r  in  the  direction  CP.  Hence 

a  =  —  co2r  cos  0  =  —  w2£ 

Since  w  is  constant  throughout  the  motion,  the  projection  is  a 
simple  harmonic  motion  in  which  c  =  co2.  Hence  any  simple  har- 
monic motion  may  be  regarded  as  a  projection  of  a  uniform  cir- 
cular motion.  The  circle  is  called  the  circle  of  reference  of  the 
simple  harmonic  motion. 

This  relation  between  the  two  kinds  of  motion  affords  a  means 
of  deducing  some  of  the  properties  of  simple  harmonic  motion 
without  the  use  of  advanced  mathematics. 

115.  Period  of  a  Simple  Harmonic  Motion. — Considei  a  simple 
harmonic  motion  as  a  projection  of  a  uniform  circular  motion. 
The  periods  of  the  two  motions  must  be  the  same.  Denote  it  by 
T.  From  §114 


-,.,2 


cozx=-  x 


Hence 


Since  x  and  a  are  always  of  opposite  signs,  the  quantity  under  the 
radical  is  always  numerically  positive. 

116.  Trigonometrical  Expression  for  the  Displacement. — From 
the  same  relation  we  can  also  deduce  an  expression  for  the  dis- 
placement at  any  moment  in  the  simple  harmonic  motion.  Let 
P  be  the  point  whose  motion  projects  into  that  of  the  vibrating 
point  M.  The  angle  PC  A  or  the  phase  of  P's  motion  (§109) 
equals  [(2v/T)t  +  e].  Hence  for  CM  or  the  ^displacement,  x,  in 
the  simple  harmonic  motion,  we  have 


While  we  have  deduced  this  expression  from  the  related  circular 
motion,  it  must  now  be  regarded  as  an  expression  for  the  simple 
harmonic  motion  of  amplitude  r  and  period  T.  [(2x/T)t  +  e]  is 
called  the  phase  of  the  simple  harmonic  motion  at  time  t,  e 
being  the  phase  of  the  simple  harmonic  motion  at  zero  time. 


82 


MECHANICS  AND  THE  PROPERTIES  OF  MATTER 


For  two  particular  values  of  e  the  expression  for  x  becomes 
simpler.  If  e  is  zero,  which,  as  we  see  from  the  circular  motion, 
means  that  at  zero  time  M  is  at  A.  the  expression  for  x  is 


x  —  r  cos 


If  e=-(7T/2),  at  zero  time  P  is  at  B 
and  M  is  therefore  at  C  and  moving  in 
the  positive  direction.  Substituting  this 
value  of  e  in  the  above  general  expression 
for  x,  we  get 


117.  Simple  Pendulum. — A  simple  pendulum  consists  of  a 
small  heavy  body,  called  the  bob  (usually  spherical),  suspended 
by  a  practically  inextensible  cord,  the  mass  of  which  is  so  small 
as  to  be  negligible  compared  with  the  bob.  As  the  pendulum 
swings  through  a  small  angle,  the  bob  vibrates  through  a  small  arc 
of  a  circle  which  is  very  nearly  a  straight  line. 

The  force  of  gravity,  mg,  on  the  bob  of  the  pendulum  acts 
vertically,  and  it  may  be  resolved  into  a  component  along  the 
tangent  and  a  component  along  the  radius.  The  latter  component 
produces  a  tension  on  the  cord  which  does  not  affect 
the  motion,  while  the  former  component  produces 
an  acceleration  along  the  tangent.  When  the  cord 
is  at  an  inclination  d  to  the  vertical,  the  com- 
ponent along  the  tangent  equals  mg  cos  [(rc/2)  —  6] 
or  mg  sin  0.  Since  the  pendulum  is  supposed  to 
vibrate  through  a  very  small  angle,  sin  0  may  be 
replaced  by  d;  in  fact,  for  values  of  6  less  than  2°, 
sin  d  and  6  are  equal  within  one  part  in  10,000. 
If  the  distance  of  the  bob  from  its  lowest  point, 
measured  along  the  tangent,  be  denoted  by  x  and 
the  length  of  the  pendulum  by  I,  0  =  x/l  radians. 
Hence  the  force  along  the  tangent  is  mg(x/l). 
This  force  is  in  the  negative  direction  when  x  is  positive. 
Hence,  denoting  the  acceleration  along  the  tangent  by  a,  we 
have  by  the  Second  Law  of  Motion 


nig 

Fio.  59. — Simple 
pendulum. 


KINEMATICS  AND  DYNAMICS  83 

Hence  fl £x 

Since  the  multiplier  of  a;  is  a  constant,  the  acceleration  is 
opposite  to  and  proportional  to  the  displacement.  Hence  the 
motion  is  simple  harmonic  motion,  and,  if  T  be  the  period  or 
time  of  vibration  of  the  pendulum,  by  §115 


118.  Angular  Harmonic  Motion. — A  body  attached  to  an  axis 
may  vibrate  backward  and  forward  through  an  angle,  as  in  the 
case  of  a  balance  wheel  of  a  watch  or  of  any  heavy  body  hung  on 
a  peg.  When  the  angular  acceleration,  a,  is  always  opposite  to 
and  proportional  to  the  angular  displacement,  6,  the  motion  is 
called  angular  harmonic  motion.  Hence  the  general  formula  for 

such  motion  is 

a=-C-d 
C  being  a  constant. 

Let  Fig.  60  be  a  plane  through  the  body 
perpendicular  to  the  axis  0.  A  line  OM  in 
the  body  will  vibrate  backward  and  forward 
through  an  angle.  The  point  M  will  vibrate 
in  an  arc  of  a  circle  of  radius  OM  or  r.  When 
the  angular  displacement  of  OM  from  its  mean 
position  OC  is  0,  the  displacement,  x,  of  M 
from  C  is  rO  and  the  linear  acceleration,  a,  of 
M  is  ra  (§69).  Substituting  these  values 
of  6  and  a  in  the  above  formula  and  cancelling  r,  we  get 

a=-C-x 

Thus  the  motion  of  M  is  simple  harmonic  motion  in  all  respects 
except  that  it  is  along  an  arc  (which  may  be  long  or  short)  instead 
of  along  a  straight  line.  We  might  suppose  the  arc  straightened 
out  without  any  other  change  in  the  nature  of  the  motion  of 
M.  Hence,  if  T  be  the  period  of  M's  motion,  which,  of  course,  is 
the  same  as  the  period  of  the  angular  harmonic  motion, 


84  MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

This  expression  for  the  calculation  of  the  period  of  an  angular 
harmonic  motion  is  similar  to  that  for  the  calculation  of  the 
period  of  a  simple  harmonic  motion  (§115). 

As  examples  of  angular  harmonic  motion  we  shall  consider 
the  torsion    endulum  and  the  physical  pendulum. 

.,  119.  The  Torsion  Pendulum.  —  A  torsion  pendu- 
lum  consists  of  a  vertical  wire  carrying  a  body  at 
one  end  and  clamped  at  the  other  end.  When  the 
body  is  turned  around  the  wire  as  axis  and  re- 
leased it  performs  angular  vibrations;  the  twisted 
wire  begins  to  untwist  and  thus  starts  the  motion 
which  persists  after  the  wire  has  untwisted,  owing 
to  the  kinetic  energy  acquired  by  the  body. 
el.—  To  twist  the  wire  requires  the  application  of  a 
couple-  The  twist,  6,  produced  by  a  certain  couple 
of  moment  L,  is  proportional  to  L  and  to  the  length 
I  of  the  wire.  Hence  LI  =  -id,  where  T  is  a  constant  which  is 
called  the  constant  of  torsion  of  the  wire.  The  couple  exerted 
by  the  twisted  wire  is  equal  and  opposite  to  that  required  to 
produce  the  twist.  Hence  the  couple  exerted  by  the  wire  on 
the  body  is  —  r(0/J)  when  the  displacement  is  0.  This  couple 
gives  the  body  an  angular  acceleration,  and,  if  we  denote  this 
by  a  and  the  moment  of  inertia  of  the  body  by  /, 

.4.,. 


In  this  the  multiplier  of  6  is  a  constant  which  depends  on  the  wire 
and  the  body  and  is  independent  of  the  motion.  Hence  the 
motion  agrees  with  the  definition  of  angular  harmonic  motion, 
and,  if  T  is  the  period  of  vibration, 


It  should  be  noticed  that  we  have  not  assumed  the  angle  of 
vibration  to  be  small,  as  in  the  case  of  the  ordinary  pendulum;  in 
the  torsion  pendulum  the  restoring  couple  is  proportional  to 


KINEMATICS  AND  DYNAMICS 


85 


the  angular   displacement,  even  when  the  latter  is  large  (pro- 
vided it  is  not  so  large  as  to  permanently  strain  the  wire)  . 

By  means  of  the  torsion  pendulum  the  moment  of  inertia  of  an  irregular 
body  can  be  compared  with  that  of  a  body  of  known  moment  of  inertia. 
The  two  are,  by  the  above  formula  for  T,  proportional  to  the  squares 
of  the  corresponding  times  of  vibration  when  the  bodies  are  in  turn 
attached  to  the  same  wire  and  set  into  angular  vibration. 

120.  The  Compound  Pendulum.  —  A  body  of  any  shape  sus- 
pended by  a  horizontal  axis  and  vibrating  under  gravity  through 
a  small  angle  constitutes  a  compound  pendulum.  Fig.  62  repre- 
sents a  vertical  section  through  the  center  of  gravity  C  and 
perpendicular  to  the  axis  of  suspension  £.  Denote  SC  by  h. 
When  SC  is  inclined  at  an  angle  0  to  the  vertical, 
the  force  of  gravity,  mg,  which  acts  at  C,  has  a 
moment  about  S  equal  to  mgh  sin  0,  which  is  nega- 
tive when  0  is  positive.  Hence,  if  I  is  the  moment 
of  inertia  of  the  body  about  S, 

—  mgh  am  6  =  la 

If  the  angle  6  is  always  small,  we  may,  as  in  th  e 
case  of  the  simple  pendulum,  replace  sin  6  by  6 
and  thus  get 


This  satisfies  the  condition  for  angular  harmonic  motion  and  the 
period  of  vibration  is 


mgh 

Let  the  radius  of  gyration  about  an  axis  through  C  parallel  to  the  axis 
of  suspension  be  k.  Then  the  moment  of  inertia  about  the  axis  through 
C  is  mfca  and  about  the  parallel  axis  through  S  it  is  mk*  +  mh*  (§§83,  85), 
which  is,  therefore,  the  value  of  /.  Hence 

]W+h* 

gh 

If  this  be  compared  with  the  formula  for  a  simple  pendulum,  it  is  seen  that, 
if  I  be  the  length  of  a  simple  pendulum  that  vibrates  in  the  same  time  as 
the  compound  pendulum, 

t* 


86 


MECHANICS  AND  THE  PROPERTIES  OF  MATTER 


V//7///////////////7///////A 


Hence  (l-h)h-k* 

The  length  I  is  evidently  greater  than  h.  Hence,  if  we  measure  along  SC 
a  length  equal  to  Z,  we  shall  arrive  at  a  point  0  in  SC  extended.  The 
point  0,  which  is  always  on  the  opposite  side  of  C  from  S,  is  the  point 
at  which  the  whole  mass  of  the  body  might  be  supposed  concentrated  with- 
out any  alteration  of  the  period  of  vibration.  0  is  called  the  center  of 
oscillation  corresponding  to  the  axis  of  suspension  S.  Since  C0=-  (I— h) 
and  CS  =  h,  we  have  as  the  relation  between  any  center  of  oscillation  and 
the  position  of  the  corresponding  axis  of  suspension 

OS-CO-*' 

If  the  pendulum  be  now  inverted  and  set  to  vibrate  about  an  axis  through 
0  parallel  to  the  former  axis,  the  new  center  of  oscillation,  0',  will  lie  in 
OC  produced  and  must  satisfy  the  relation 

CO-CO'  -*> 

A  comparison  of  these  two  equations  shows  that  O*  must  coincide  with  S. 
Hence  the  center  of  suspension  and  the  center  of  oscillation  are  interchangeable 
and  the  distance  between  them  is  the  length  of  the  equivalent  simple  pendulum. 
This  is  the  principle  of  Kater's  pendulum. 

121.  Energy  Changes. — The  resultant  force  of  gravity  acts  at  C  (Fig. 

62).  Hence  the  potential  energy  of  the  pendulum  in  any 
position  is  the  same  as  if  its  mass  were  concentrated  at  C. 
But  the  pendulum  does  not  swing  as  if  it  were  concen- 
trated at  C,  because  its  kinetic  energy  is  that  of  its  mass 
supposed  concentrated  at  C  plus  its  kinetic  energy  of 
rotation  about  C  (§91).  As  the  pendulum 
falls  toward  the  vertical  the  lost  potential 
energy  goes  partly  into  energy  of  rotation 
about  C;  hence  it  does  not  swing  as  rapidly 
as  if  it  were  concentrated  at  C,  that  is,  as  if  it 
were  a  simple  pendulum  of  length  SC.  A  parallel  case  that 
brings  out  the  distinction  is  illustrated  by  a  block  suspended 
by  two  cords  as  in  Fig.  63.  Swinging  perpendicularly  to  the 
plane  of  the  figure  it  is  a  physical  pendulum  of  length  SO, 
the  block  having  energy  of  rotation.  Swinging  parallel  to 
the  plane  of  the  figure  it  is  a  simple  pendulum  of  length  equal 
to  the  length  of  the  cords;  the  block  in  this  case  has  no  rota- 
tion. A  similar  explanation  applies  to  the  motion  of  the 
pans  of  a  balance.  They  do  not  rotate  with  the  beam  but 
move  vertically;  hence  they  affect  the  motion  as  if  concen- 
trated on  the  supporting  knife-edges. 

122.  Center  of  Percussion.    There  is  another  important  re- 
lation between  an  axis  of  suspension  S  and  the  corresponding 

center  of  oscillation  0.  A  blow  at  0  transverse  to  SO  will  start  the  body 
rotating  about  S  without  any  jar  on  the  support  at  S.  Hence  0  is  also 
called  the  center  of  percussion  of  the  body  when  suspended  at  S.  The 
center  of  percussion  is  readily  found  by  suspending  the  body  by  a  cord 


FIG.  63. 


FIG.  64.— 
Center  of 
percussion. 


KINEMATICS  AND  DYNAMICS 


87 


and  striking  horizontal  blows  at  various  points.  Or  it  may  be  found  by 
holding  the  body  at  S  and  striking  across  a  table  edge,  as  a  base-ball 
player  strikes  a  ball  with  a  bat;  when  the  blow  is  through  the  center  of 
percussion  there  is  no  jar  on  the  hand. 

123.  Gyroscopic  Motion. — A  gyroscope  is  a  wheel  on  a  hori- 
zontal axle  which  is  supported  on  a  pivot  (a  bicycle  wheel  sus- 
pended by  a  vertical  cord  attached  to  a  short  extension  of  the 
axle  will  serve).  When  the  wheel  is  set  in  rotation  and  the 
axle  then  released,  the  axle,  instead  of  tilting  in  a  vertical 
plane,  as  it  would  if  the  wheel  were  at  rest,  revolves  in  a  hori- 
zontal plane  at  a  rate  that  depends  on  the  velocity  of  rotation 
of  the  wheel  about  the  axle.  This  motion  is  called  precession. 
(Slight  vertical  oscillations  or  nutations  of  the  free  end  of  the 
axle  may  also  accompany  the  precession.)  The  weight  of  the 


Fia.  65. — A  gyroscope. 


Fio.  66. 


wheel  acting  at  the  center  of  the  wheel  has  a  moment  about  an 
axis  through  the  pivot  at  right  angles  to  the  axis  of  the  wheel. 
If  this  moment  of  force  be  increased  by  hanging  a  weight  on 
the  frame  the  rate  of  precession  will  be  greater.  If  the  wheel 
be  supported  at  its  center  of  gravity  there  will  be  no  moment 
of  force  and  no  precession.  (Thus  mounted  the  instrument  is 
sometimes  called  a  gyrostat.) 

If  the  motion  be  carefully  considered  it  will  be  seen  that  it  is 
very  analogous  to  the  revolution  of  a  particle  in  a  circle  under 
the  action  of  a  force  directed  toward  the  center.  The  latter 
requires  a  force  perpendicular  to  the  direction  of  motion^  while 
precession  requires  a  moment  of  force  about  an  axis  perpendicular 
to  the  axis  of  rotation. 

124.  Moment  of  Force  Required  to  Produce  Precession. — To  find  the 
magnitude,  L,  of  the  mordent  of  force  required  for  precession,  let  OA 


88          MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

represent  the  angular  momentum,  Ia>.  After  a  short  time,  t,  OA  will  have 
turned  through  a  small  angle  <f>  to  the  position  OBt  where  <j>—<t>'tt  at'  being 
the  angular  velocity  of  precession.  Hence  angular  momentum  represented 
by  OZ  or  AB  must  have  been  added  and  this  must  equal  Lt  (§87)- 

.  ^»M».    / 

Hence  L^Iata)' 

Whenever  a  rotating  body  shows  precession,  that  is,  when  its  axis  of  rota- 
tion is  revolving,  some  agent  must  be  applying  the  moment  of  force,  L, 
about  an  axis  perpendicular  to  those  of  rotation  and  precession  and  must 
be  experiencing  an  equal  and  opposite  reaction. 

125.  Other  Examples  of  Precession. — The  curvature  of  the  path 
of  a  coin  rolled  with  a  tilt  along  a  table  is  due  to  the  precession  of 
its  axis  caused  by  the  moment  of  its  weight  about  the  point  of 
contact  with  the  table.     The  motion  of  a  top  is  a  precession. 

Any  large  body,  such  as  a  dynamo  armature,  in  rotation  aboard 
a  vessel  that  is  rolling,  pitching,  or  turning,  has  a  precessional 
motion  and  the  bearings  must  supply  the  necessary  moment  of 
force  and  experience  an  equal  and  opposite  reaction. 

When  a  side- wheel  steamer  is  turned  in  a  sharp  curve  there  is  a 
precession  of  the  axis  of  the  paddle  wheels.  To  produce  this 
precession  and  at  the  same  time  keep  the  vessel  level  would 
require  a  moment  of  force  about  a  longitudinal  axis,  and  in  the 
absence  of  such  a  moment  the  vessel  lists  to  the  outer  side. 

The  earth  is  not  quite  spherical  but  bulges  at  the  equator.  On 
one  side  the  protuberance  is  closer  to  the  moon  than  the  center  of 
the  earth  and  on  the  other  side  it  is  farther  away.  The  result  of 
this  (and  of  a  similar  but  smaller  moment  exerted  by  the  sun)  is 
a  moment  of  force  that  causes  a  precession  of  the  earth's  axis. 

The  gyroscope  has  been  applied  to  steering  torpedoes,  to  pre- 
venting the  rolling  of  ships,  to  balancing  trains  on  a  single  rail, 
and  in  the  construction  of  a  non-magnetic  mariner's  compass. 

• 

FRICTION 

126.  Static  Friction. — When  two  solids  are  in  contact  there  is  a 
resistance,  caused  by  the  surfaces,  to  the  sliding  of  one  on  the 
other.     This  resistance  is  called  Friction.     When  a  force  parallel 
to  the  surfaces  of  contact  is  applied  to  one  of  the  bodies  and  the 


FRICTION  89 

force  is  less  than  a  certain  amount,  which  depends  on  the  nature 
of  the  surfaces  and  the  pressure  between  them,  motion  will  not 
take  place,  the  resistance  being  equal  to  the  force.  When  the 
force  is  increased  to  a  certain  value  the  resistance  will  fail  to 
increase  and  sliding  will  take  place.  This  maximum  resistance 
is  called  the  maximum  static  friction.  With  a 
given  pair  of  surfaces  in  contact  and  with  a  »^]j§mm 
force  tending  to  produce  sliding  motion  in  a  j; 

certain  direction  (to  take  account  of  the  influ-  Fia.  57. 

ence  of  grain),  the  maximum  static  friction  is 
found  to  be  (within  certain  wide  limits  of  pressure)  proportional 
to  the  pressure.     Denoting  the  coefficient  of  proportionality  by 
//,  the  maximum  of  static  friction  by  F,  and  the  pressure  by  P, 
we  have 


The  constant  /j.  is  called  the  coefficient  of  static  friction,  and  may  be 
denned  as  the  ratio  of  the  maximum  static  friction  between  two 
surfaces  to  the  pressure  between  them. 

By  the  pressure  here  is  meant  the  total  perpendicular  force 
between  the  surfaces  (not  the  force  per  unit  of  area,  as  the  word 
pressure  is  sometimes  used).  If  one  of  the  two  bodies  rests  on 
the  other  and  if  the  surfaces  of  contact  are  plane  and  horizontal, 
the  pressure  is  the  weight  of  the  upper.  The  maximum  static 
friction  is  the  force  applied  horizontally  to  the  upper  that  will  just 
produce  motion.  If  additional  weights  be  placed  on  the  upper 
body,  the  pressure  between  the  surfaces  will  be  increased  and  the 
friction  will  be  increased  in  the  same  proportion.  If  the  upper 
mass  be  redistributed  in  any  way,  for  instance,  if  it  be  cut  in  two 
and  one  part  placed  on  the  other,  the  total  force  of  friction  will 
not  change;  for,  while  the  area  of  contact  will  be  diminished,  the 
pressure  on  each  unit  of  area  will  be  increased  in  the  same  pro- 
portion. Thus  the  total  frictional  resistance  is  independent  of  the 
area  of  contact  and  for  two  given  surfaces  depends  only  on  the 
pressure,  as  is  implied  in  the  equation  F=/iP. 

127.  The  coefficient  of  static  friction  between  two  surfaces  depends  on  the 
materials  and  a  variety  of  circumstances.  The  rougher  the  surfaces,  that 
is  the  greater  the  inequalities  in  each,  the  larger  is  p.  If  the  surfaces  are 
not  clean  parts  of  the  surfaces  are  replaced  by  surfaces  of  the  foreign 


90          MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

substance  and  p  is  necessarily  different.  The  longer  two  surfaces  are  in 
contact  the  greater  the  maximum  static  friction;  this  is  especially  true 
of  soft  or  fibrous  surfaces.  When  the  materials  are  of  grained  structure 
the  friction  is  greater  across  the  grain  than  along  it.  Friction  is,  no  doubt, 
due  to  interlocking  of  the  projections  on  one  surface  with  those  on  the 
other  surface.  When  slipping  takes  place  some  projecting  pieces  are 
broken  off  or  abraded  as  it  is  called.  With  prolonged  contact  between  two 
surfaces  small  readjustments  of  the  surface  particles  take  place,  so  that 
the  fit  becomes  closer  and  the  resistance  to  motion  greater.  It  has  even 
been  found  that  when  one  surface  is  pushed  a  very  small  distance  it  will 
when  released  spring  back,  thus  showing  that  there  is  some  elastic  bending 
of  surf  ace  projections.  In  general,  friction  between  two  surfaces  of  the 
same  material  is  greater  than  between  surfaces  of  different  material  since 
the  former  allows  more  uniform  interlocking.  Thus  there  is  an  advantage 
in  using  brass  bearings  for  steel  shafts  to  diminish  friction,  and  covering 
with  leather  the  face  of  a  pulley  used  with  leather  belting  increases  friction 
and  helps  to  prevent  slip. 

Friction  is  utilized  in  the  transmission  of  energy  by  machine  belting. 
Usually  some  slipping  takes  place,  for  the  belt  stretches  somewhat  while 
in  contact  with  the  pulley.  Friction  between  the  driving  wheels  of  a 
locomotive  and  the  rails  prevent  slipping;  without  it  the  locomotive  would 
be  helpless,  and  where  it  is  not  sufficient  the  track  is  sanded.  To  hold  a 
rope  fast  it  is  sometimes  wrapped  around  a  post.  The  friction  on  each 
part  of  the  rope  diminishes  the  tension  transmitted  to  the  next  part.  It 
is  found  that  after  one  turn  the  tension  is  diminished  to  about  -J-,  after 
two  turns  to  ^  of  ^  and  so  on.  At  this  rate  after  five  turns  a  pull  of  one 
pound  weight  on  the  free  end  would  counteract  a  force  of  4  tons  at  the 
other  end. 

The  laws  of  friction  were  first  investigated  by  Coulomb  and  are  some- 
times called  by  his  name. 

128.  Slip  on  an  Incline. — When  a  body  rests  on  an  inclined 
plane  the  tilt  of  which  is  gradually  increased 
there  is  some  angle  i  at  which  slipping  begins. 
The  weight  of  the  body  is  mg  and  acts 
vertically.     It  may  be  resolved  into  a  com- 
ponent mg  sin  i  down  the  plane  and  a  compo- 
nent mg  cos  i  perpendicular  to  the  plane. 
The  latter  component  causes  pressure  between 
FIO  es  tne  surfaces,  while  the  former  is  the  force  par- 

llel  to  the  surface  which  produces  motion. 
Hence  from  the  definition  of  this  coefficient  of  static  friction 

F  mg  sin  i 
/£  =  =  =—  — ^ 
^  P  mg  cos  i 


FRICTION  91 

Thus  the  coefficient  of  static  friction  is  equal  to  the  tangent  of  the 
angle  of  slip.  (This  angle  is  also  sometimes  called  the  angle  of 
repose.)  This  relation  provides  a  simple  method  of  measuring  //. 
129.  Kinetic  Friction.  —  To  keep  one  body  sliding  on  another  at 
a  constant  speed  a  certain  force,  F,  parallel  to  the  surface  of 
contact  is  required.  Through  a  considerable  range  of  speed  this 
force  is  practically  constant.  The  opposing  resistance  offered  by 
the  surfaces  is  called  kinetic  friction.  It  is  found  to  be,  for  a 
certain  pair  of  surfaces  moving  in  a  definite  direction,  proportional 
to  the  pressure,  P,  between  the  surfaces.  Denoting  the  coeffi- 
cient of  proportionality  by  //,  we  have 


The  constant  //  is  called  the  coefficient  of  kinetic  friction.  It  may 
be  defined  as  the  ratio  of  the  kinetic  friction  between  two  surfaces 
to  the  pressure  between  them. 

As  in  the  case  of  static  friction,  for  a  given  pressure  between 
two  surfaces  the  kinetic  friction  is  independent  of  the  area  of 
contact. 

The  kinetic  friction  between  two  surfaces  is  in  general  less 
than  the  maximum  static  friction.  The  reason  probably  is  that 
time  is  not  allowed  for  the  surface  to  settle  into  as  close  contact 
as  if  they  were  at  rest.  Moreover,  kinetic  friction  is  not  quite 
independent  of  velocity.  When  the  velocity  is  decreased  until  it 
is  very  small  (how  small  depends  upon  the  particular  surfaces) 
the  friction  increases  and  it  continues  to  increase  as  the  velocity 
diminishes  toward  zero,  and  at  a  sufficiently  small  velocity  the 
kinetic  friction  probably  does  not  differ  appreciably  from  the 
maximum  static  friction.  At  very  great  velocities  the  friction  is 
generally  less  than  at  moderate  velocities. 

A  friction  dynamometer  is  a  machine  for  measuring  the  power  of  an 
engine;  the  engine  drives  a  wheel  over  which  a  belt  hangs  under  known 
tension.  From  the  tension  of  the  belt  and  the  number  of  revolutions 
made  by  the  latter  the  work  done  is  calculated. 

When  a  lubricant  is  used  between  two  surfaces  there  is  no  longer  friction 
of  solid  on  solid  and  the  laws  of  kinetic  friction  no  longer  hold;  the  coeffi- 
cient of  friction  depends  on  both  pressure  and  velocity  and  the  action 
is  very  complex.  The  friction  of  a  skate  on  ice  is  probably  greatly  di- 
minished by  the  momentary  liquefaction  of  the  ice  immediately  under  the 


92          MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

skate  due  to  the  great  pressure  exerted  by  the  latter  on  a  small  area  (see 
§301"). 

130.  Sliding  on  an  Inclined  Plane. — A  body  sliding  down  an  in- 
clined plane  (Fig.  21)  is  urged  downward  by  the  component  of 
its  weight  along  the  plane  and  retarded  by  friction.     If  the  in- 
clination of  the  plane  to  the  horizontal  is  it  the  component  of 
gravity  along  the  plane  is  mg  sin  i.     The  pressure  perpendicular 
to  the  plane  is  mg  cos  i;  hence  the  force  of  friction  is  //  mg  cos  i. 
If  the  component  of  gravity  down  the  plane  exceeds  the  force 
of  friction,  the  body  will  slide  with  an  acceleration  a.     Hence, 
taking  the  direction  down  along  the  plane  as  the  positive  direction, 
we  have  by  Newton's  Second  Law 

ma  =  mg  sin  i — p'mg  cos  i 

This  suggests  a  method  of  finding  //  by  measuring  a  and  i. 

131.  Rolling  Friction. — The  term  friction  is  also  applied  to  the 
resistance  experienced  by  a  wheel  in  rolling  on  a  surface  without 
any  slipping.     The  cause  of  the  resistance  is  in  this  case  entirely 
different.     This  is  seen  by  considering  the  rolling  of  a  heavy 

wheel  on  a  soft  substance,  such  as  India  rub- 
ber. If  the  wheel  were  at  rest  it  would  sink 
into  the  rubber,  raising  a  small  mound  on  each 
side  of  the  contact.  When  the  wheel  is  mov- 
ing forward  the  mound  is  chiefly  on  the  for- 
ward side  at  A.  The  pressure,  P,  of  the  rub- 

Fio    69. — Resistance        ,  ,-,  ,        ,  *      •       •       T        i     ,         ,1 

to  rolling.  ^er   on   the  wheel   at  A  is  inclined  to  the 

vertical,  in  some  such  direction  as  AP.     The 

point  about  which  the  wheel  is  momentarily  rotating  (§69)  is 

not  C  but  B  in  the  figure,  and  the  moment  of  P  about  B  is 

necessarily  opposed  to  that  of  F. 

It  will  be  noted  by  the  explanation  that  the  resistance  to  the  motion  is 
greater  the  softer  the  surface,  greater  the  greater  the  pressure  of  the  wheel 
on  the  surface,  and  less  the  larger  the  wheel,  since  a  larger  wheel  will  dis- 
tribute its  pressure  over  a  larger  surface  and  will  not  sink  so  deeply.  When 
the  surface  on  which  the  wheel  rolls  is  hard  very  little  deformation  will 
ensue,  and  the  resistance  to  the  motion  will  be  much  less.  Thus  the 
resistance  to  the  rolling  of  iron  on  india  rubber  is  about  ten  times  greater 
than  the  rolling  of  iron  on  iron.  With  a  lignum  vitse  cylinder  of  16-in. 
diameter  loaded  with  1000  Ibs.  the  rolling  friction  has  been  found  to  be 


SIMPLE  MACHINES  93 

about  3  per  cent,  of  the  sliding  friction  when  the  wheel  was  not  allowed 
to  rotate.  Because  of  this  difference  rolling  is,  when  possible,  preferred 
to  sliding.  Thus  rollers  beneath  a  heavy  body  and  the  balls  in  a  ball- 
bearing greatly  diminish  fractional  resistance. 

A  pneumatic  tire  on  a  bicycle  or  automobile  flattens  out  in  contact  with 
the  ground  and  does  not  sink  in,  so  that  it  gives  the  wheel  the  advantage 
of  a  much  larger  wheel.  But  it  also  bulges  a  little  in  front  of  the  flattened 
part  and  this  bulge  is  an  obstruction  of  the  same  nature  as  the  little  mound 
in  Fig.  69.  On  a  perfectly  smooth,  plane,  hard  road  a  pneumatic  tire  would 
be  a  disadvantage.  On  a  soft  rough  road  it  is  a  great  advantage.  For 
a  hard  smooth  road  the  tire  should  be  pumped  "hard"  for  a  soft  road  it 
should  be  "soft." 

SIMPLE  MACHINES 

132.  Machines. — A  machine  is   a  contrivance  for  applying 
energy  to  do  work  in  the  way  most  suitable  for  a  certain  purpose. 
The  machine  does  not  create  energy;  no  machine  can  do  that. 
To  do  work  it  must  receive  energy  from  some  store  of  energy,  and 
the  greatest  amount  of  useful  work  it  can  do  cannot  exceed  the 
energy  it  receives. 

Different  machines  receive  energy  in  different  forms,  some  in 
the  form  of  mechanical  (kinetic  and  potential)  energy,  some  in 
the  form  of  heat  energy,  some  in  the  form  of  chemical  energy,  and* 
so  on.     We  shall  only  consider  here  machines  which  employ 
mechanical  energy  and  do  work  against  mechanical  forces. 

In  certain  very  elementary  machines,  the  so-called  simple 
machines,  the  agent  which  supplies  the  energy  exerts  but  a  single 
force  and  the  machine,  at  least  as  regards  the  useful  work  which 
it  performs,  is  opposed  by  a  single  resisting  force.  The  former  is 
frequently  called  the  "  power ";  but,  to  avoid  confusion,  we  shall 
call  it  the  applied  force.  The  resisting  force  is  frequently  called 
the  "weight";  but,  as  the  opposing  force  is  not  always  that  of 
gravity,  we  shall  call  it  the  resistance. 

Every  machine  in  its  action  encounters  a  certain  amount  of 
frictional  resistance;  the  work  done  against  it  is  not  usually  useful 
work.  This  in  many  cases  is  very  small,  and,  in  treating  (to  a 
first  approximation)  of  the  simple  machines,  it  is  customary  to 
neglect  it. 

133.  Mechanical  Advantage. — The  work  done  by  the  applied 
force,  P,  is  measured  by  the  product  of  P  and  the  distance,  p, 


94          MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

through  which  P  acts.  The  work  done  against  the  resistance  is 
measured  by  the  product  of  the  resistance,  Q,  and  the  distance 
q,  through  which  it  is  overcome.  In  a  simple  machine  (where 
friction  may  be  neglected)  these  must  be  equal.  Hence 

Q_p 


Hence  p  is  greater  than  q  in  the  proportion  in  which  Q  is  greater 
than  P.  This  principle  was  first  stated  by  Stevinus  (1548-1620)  . 
It  is  frequently  put  in  the  form  "what  is  gained  in  power  (i.e. 
force)  is  lost  in  speed." 

The  ratio  of  Q  to  P  for  a  machine  is  called  the  mechanical 
advantage  of  the  machine.  Since,  for  a  perfect  machine,  that  is, 
one  in  which  friction  is  negligible,  the  above  ratio  is  also  the 
ratio  of  p  to  g,  it  follows  that  we  can  deduce  the  mechanical 
advantage  of  such  a  machine  from  the  ratio  of  the  speeds  without 
considering  the  inner  mechanism  of  the  machine. 

134.  Efficiency.  —  By  the  efficiency  of  a  machine  is  meant  the 
ratio  of  the  useful  work,  or  work  of  the  kind  desired,  to  the  energy 
received.    For  a  simple  machine  without  friction  this  would  be 
unity.     When  there  is  friction  the  efficiency  may  have  any  value 

"less  than  unity. 

135.  Levers.  —  A  lever  is  a  bar  supported  at  a  point  called  the 
fulcrum,  F;  a  force,  P,  applied  to  the  bar  at  a  point  A  will  over- 


f      i 

*k.  F 

> 
A        F  _,. 

Q 

6,__ 

P 

1- 

B-~~^ 

^•M.^^ 

I 

B 
a 

ii   * 

FIG.  70.—  -Three 

classes  of  levers. 

III 


come  a  resistance,  Q,  acting  at  another  point  B.  We  shall  sup- 
pose that  P  and  Q  act  at  right  angles  to  the  bar  and  to  the  axis  of 
rotation  at  the  fulcrum. 

To  find  the  relation  between  P  and  Q  suppose  the  bar  to  turn 
through  a  very  small  angle,  so  that  A  moves  through  a  distance 
Aa  and  B  through  a  distance  Bb.  The  work  done  by  P  is  P-Aa 


SIMPLE  MACHINES  95 

and  the  work  done  against  Q  is  Q  -Bb.     The  conservation  of  energy 
requires  that  these  should  be  equal.     Hence 


_ 
P  "Bb  ~BF 

This  relation  may  also  be  found  by  considering  the  parallel 
forces  acting  on  the  bar  or  by  taking  moments  about  the 
fulcrum. 

Levers  are  usually  divided  into  three  classes  represented  by 
the  figures.  In  levers  of  the  first  class  the  force,  P,  and  the 
resistance,  Q,  are  on  opposite  sides  of  the  fulcrum,  and  the 
resistance  may  be  greater  or  less  than  the  applied  force.  To 
this  class  belong  the  crow-bar,  forceps,  scissors,  poker,  and  the 
common  balance. 

In  levers  of  the  second  class  the  applied  force  and  the  resistance 
are  on  the  same  side  of  the  fulcrum,  the  former  being  farther  from 
it  than  the  latter.  Thus  the  resistance  is  always  greater  than  the 
applied  force.  This  class  includes  the  oar  of  a  boat,  a  pair  of  nut- 
crackers, a  claw-hammer  for  extracting  nails,  etc. 

In  levers  of  the  third  class  the  applied  force  and  the  resistance 
are  on  the  same  side  of  the  fulcrum,  the  former  being  nearer  to  the 
fulcrum  than  the  latter.  The  purpose  of  such 
a  lever  is  a  gain  of  displacement  or  of  speed. 
This  class  includes  the  forearm  which  is 
hinged  at  the  elbow  and  acted  on  by  the 
biceps  at  a  distance  of  two  or  three  inches 
from  the  elbow,  a  pair  of  tongs  and  the  lever 
of  a  safety-valve  for  steam  pressure. 

136.  The  Wheel  and  Axle.—  A  straight 
lever  cannot  raise  a  weight  higher  above  the 
fulcrum  than  the  distance  of  the  weight 
from  the  fulcrum.  The  apparatus  called  a 
"  wheel  and  axle"  acts  on  the  same  principle  as  a  lever  but  its 
range  is  not  so  limited.  It  consists  of  a  wheel  of  large  radius 
rigidly  connected  to  an  axle  of  smaller  radius.  The  applied 
force,  P,  acts  on  a  cord  wrapped  around  the  wheel,  while  the 
weight  or  resistance  acts  on  a  cord  wrapped  around  the  axle. 
The  principle  involved  is  that  of  a  lever  of  the  first  class,  the 
radius,  R,  of  the  wheel  being  the  lever  arm  for  the  applied 


96 


MECHANICS  AND  THE  PROPERTIES  OF  MATTER 


force,  while  the  radius,  r,  of  the  axle  is  the  lever  arm  of  the 


resistance.     Hence 


R 

r 


This  formula  may  also  be  proved  directly  by  equating  the  work 
done  by  P  in  one  complete  revolution,  2nRP1  to  the  work  done 
against  Q,  2nrQ'f  also  by  taking  moments  about  F. 

The  principle  of  the  Wheel  and  Axle  is  ap- 
plied in  the  pilot  wheel  and  in  the  capstan 
where  the  wheel  is  replaced  by  spokes  in  the 
axle,  and  in  the  winch,  where  there  is  but  a 
single  spoke,  the  crank  arm. 

In  the  above  we  have  neglected  friction, 
which  is  always  considerable. 

137.  Differential  Wheel  and  Axle.  —  To  obtain 
a  very  high  mechanical  advantage  the  wheel 
would  have  to  be  made  very  large,  which  would 
be  inconvenient,  or  the  axle  would  have  to  be 
made  very  small,  which  would  greatly  weaken 
it.     To  avoid  these  disadvantages  the  axle  is 
made  in  two  parts  of  different  size  and  the  cord  is  wrapped  in 
the  same  direction  around  both,  as  indicated  in  the  figure,  the 
weight  being  carried  by  a  pulley  through  which  the  cord  passes. 

Let  the  radius  of  the  wheel  be  R,  that  of  the  large  part  of  the 
axle  r,  and  of  the  small  part  r'.  The  upward  force  on  the  pulley 
is  twice  the  tension  of  the  cord  and  the  downward  force  is  Q, 
the  weight  of  the  pulley  being  neglected.  Hence,  by  the  principle 
of  forces  in  equilibrium,  the  tension  in  the  cord  is  JQ.  In  one 
revolution  P  does  work  P-2xR  and  the  tension  of  the  cord  acting 
on  the  smaller  part  of  the  axle  does  work  iQ-2^r',  while  work 
iQ-27rr  is  done  against  the  tension  in  the  cord  acting  on  the  larger 
part  of  the  axle.  Hence  • 


Fio.  72-— Differential 
wheel  and  axle. 


Q_  2R 
'- 


138.  Pulleys.  —  The  simplest  pulley  is  a  wheel  for  the  purpose 
of  changing  the  direction  in  which  a  force  is  applied.  It  con- 
sists of  a  wheel  in  a  framework  or  block  which  is  either  fixed 


SIMPLE  MACHINES 


97 


or  free.     If    it  is  fixed,  the  direction   of  the  force  is  changed 
without  any  change  in  the  magnitude  (see  Fig.  73a). 

If  it  is  free  and  the  two  parts 
of  the  cord  are  parallel,  the  ten- 
sion in  any  part  of  the  cord  is 
(neglecting  friction  and  the 
weight  of  the  cord)  equal  to  the 
force  applied  at  its  free  end. 
Hence  for  equilibrium 

Fio.  73a.          Fia.  736.  ~       rt  r> 


If  the  weight  of  the  pulley  is  not  negligible  it  may  be 
included  in  Q.  This  formula  is  also  readily  found  by 
the  principle  of  energy;  for  each  unit  of  the  length  that 
Q  moves  P  must  move  two. 

139.  Block  and  Tackle. — Several  pulleys  are  fre- 
quently used  in  combination  so  as  to  secure  higher 
mechanical  advantage.  The  most  common  arrange- 
ment is  called  the  block  and  tackle.  The  pulleys  are 
in  two  blocks  with  several  pulleys  in  each  block. 
The  fixed  end  of  the  cord  may  be  attached  to  either 
the  upper  or  the  lower  block;  if  to  the  former,  there  will  be 
an  equal  number  of  pulleys  in  the  two  blocks,  as  in  the 
figure;  if  to  the  latter,  there  will  be  one  more 
pulley  in  the  upper  block.  When  the  distance 
between  the  blocks  is  decreased  by  one  unit  of 
length,  each  branch  of  the  cord  in  contact  with 
the  lower  pulley  must  shorten  one  unit  of  length. 
Hence 


Fio.  74.— 

Block  and 

tackle. 


where  n  is  the  number  of  branches  of  the  cord  at 
the  lower  block. 

140.  The  Differential  Pulley  or  Chain  Hoist. — In 
this  the  upper  block  holds  two  pulleys  of  different 
diameters  fixed  rigidly  to  the  same  axis,  while  the 
lower  block  is  replaced  by  a  single  pulley.  An 
endless  chain  passes  over  the  three  pulleys  as  shown  in  the 
figure  and  is  prevented  from  slipping  by  teeth  on  the  pulleys. 
This  is  essentially  a  modification  of  the  differential  wheel  and 


Fia.  75. 


98 


MECHANICS  AND  THE  PROPERTIES  OF  MATTER 


axle  in  which  the  wheel  and  the  larger  part  of  the  axle  have  the 
same  radius.  The  relation  between  P  and  Q,  which  may  be 
worked  out  independently  or  may  be  obtained  by  putting  R  =  r 
in  the  formula  of  §137,  is 

Q=    2r 

P    r-r* 

141.  The  Inclined  Plane. — A  force  less  than  the  weight  of  a 
body  may  suffice  to  draw  the  body  up  an  inclined  plane.  Let  P 
be  the  force  and  W  the  weight  (Fig.  76a).  Also  let  h  be  the 


height  and  I  the  length  of  the  plane.  When  the  body  has  been 
drawn  up  the  whole  length  of  the  plane  the  work  done  by  P  (ne- 
glecting friction)  will  be  PI  and  the  work  done  against  W  or  the 
increase  of  potential  energy  will  be  Wh.  These  must  be  equal. 
Hence 

E-l 

p~h 

This  is  essentially  the  same  expression  as  already  found  (§50) 
by  considering  the  component  of  W  down  the  plane.  If  friction 
cannot  be  neglected  the  work  done  against  it  will  be  Fl,  where  F 
is  the  force  of  friction,  and  in  the  above  equation  P  must  be  re- 
placed by  (P-F). 

If  P  act  horizontally  (Fig.  766)  the  work  done  by  P  will  be 
Pb.  Hence  (neglecting  friction) 

W_b 
P~h 

142.  The  Screw. — The  thread  of  an  ordinary  screw  makes  a 
constant  angle  with  the  length  of  the  screw.  If  the  thread  of  a 
vertical  screw  were  supposed  unwrapped,  with  its  inclination 
kept  constant,  it  would  be  an  inclined  line.  The  pitch  of  a  screw 


GRAVITATION  99 

is  the  distance,  parallel  to  the  length  of  the  screw,  between  con- 
secutive turns  of  the  thread.  The  pitch  divided  by  the  outer 
circumference  is  the  tangent  of  the  inclination  of  the  thread  to 
the  length  of  the  screw. 

If  a  nut  carrying  a  heavy  weight  be  turned 
around  a  vertical  screw  so  that  it  ascends,  the 
process  will  be  similar  to  forcing  a  heavy  body 
up  an  inclined  plane  by  a  horizontal  force. 
In  the  jackscrew  for  raising  heavy  bodies  the 
nut  is  fixed  while  the  screw  is  turned  by  a 
lever.      The  useful  work  performed  by  the 
screw  in  one  turn  is  the  product  of  the  resist- 
ance it  overcomes,  Q,  and  the  rise  in  one  turn, 
which  is  the  pitch  h.     The  work  done  in  the     FIO.  77.— Jackscrew. 
same  time  is  the  product  of  the  applied  force, 
P,  and  the  circumference,  2xR,  of  the  end  of  the  lever  arm. 
Equating  these  would  give  us  a  relation  between  P  and  Q ;  but 
friction  is  in  general  so  large  as  to  render  the  relation  inapplicable. 


GRAVITATION 

143.  Law  of  Universal  Gravitation. — Until  the  time  of  Newton 
the  weight  of  a  body,  or  the  measure  of  its  tendency  to  fall  to  the 
earth,  was  generally  regarded  as  an  inherent  property  of  matter 
that  needed  no  further  explanation.  To  Newton  (and  to  some  of 
his  contemporaries)  it  occurred  that  the  weight  of  a  body  on  the 
surface  of  the  earth  is  due  to  a  force  of  attraction  between  the 
body  and  the  earth,  and  that  this  attraction  is  only  a  particular 
case  of  a  universal  attraction  between  all  bodies  no  matter  where 
situated.  Newton  then  sought  to  discover  the  law  that  such  a 
force  would  have  to  follow  to  account  for  the  facts,  how  it  would 
have  to  depend  on  the  masses  of  the  bodies  and  their  distance 
apart.  Now  it  was  not  possible  for  him  to  change  the  distance 
between  a  body  and  the  center  of  the  earth  by  any  except  an  ex- 
ceedingly small  fraction,  and  the  force  between  two  bodies  of 
ordinary  size  on  the  surface  of  the  earth  was  so  small  that  it 
escaped  detection  until  a  much  later  date.  Hence  he  turned  his 
attention  to  the  motion  of  the  moon  and  the  planets. 


100        MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

Before  the  time  of  Newton,  Kepler  had,  by  a  very  extensive 
and  painstaking  study  of  the  motions  of  the  planets,  arrived  at 
certain  laws  known  as  Kepler's  Laws.  These  may  be  stated  as 
follows: 

1.  The  areas  swept  over  by  a  line  joining  a  planet  to  the  sun  are 
proportional  to  the  times. 

2.  Each  planet  moves  in  an  ellipse  in  one  focus  of  which  the 
sun  is  situated. 

3.  The  squares  of  the  periods  of  revolution  of  the  planets  are 
proportional  to  the  cubes  of  the  major  axes  of  the  ellipses. 

From  these  laws  Newton  showed  that  the  motions  of  the  planets 
could  be  accounted  for  on  the  supposition  that  between  each 
planet  and  the  sun  there  is  a  force  of  attraction,  proportional  to 
the  product  of  the  masses  and  inversely  as  the  squares  of  their 
distances  apart. 

Newton  also  showed  that,  if  we  suppose  that  there  is  a  force 
according  to  this  law  between  every  two  particles,  a  sphere  that 
is  either  homogeneous  or  may  be  regarded  as  made  up  of  shells 
each  of  which  is  ty>mogeneous  will  attract  an  outside  body  as  if 
the  sphere  were  concentrated  at  its  center.  The  earth  is  very 
nearly  such  a  sphere  and  must,  therefore,  according  to  the  law 
of  gravitation,  attract  (approximately)  as  if  concentrated  at  its 
center. 

144.  Motion  of  the  Moon. — As  evidence  for  the  law  of  gravita- 
tion Newton  showed  that  it  correctly  accounts  for  the  motion  of 
the  moon.  At  the  surface  of  the  earth  a  body  is  attracted  by  the 
earth  as  if  the  latter  were  concentrated  at  its  center.  Now  the 
radius  of  the  earth  is  approximately  4,000  miles  and  the  average 
value  of  the  acceleration  of  a  falling  body  may  be  taken  as  32.2 
feet  per  sec3.  The  distance  of  the  moon  from  the  earth,  which  is 
somewhat  variable,  may  be  taken  as  approximately  240,000 
miles  or  60  times  the  radius  of  the  earth.  Hence,  according  to 
the  law  of  gravitation,  the  acceleration  of  a  body  at  the  distance 
of  the  moon  due  to  the  earth's  attraction  should  be  32.2/603  or 
.00894  ft.  per  sec*. 

The  acceleration  a  of  the  moon  towards  the  earth  (§32)  equals 
v*/R.  The  period  of  rotation  of  the  moon,  also  slightly  variable, 
is  about  27  days,  8  hours.  Calling  this  T,  we  have  v  =  (2nR/T). 
Hence  a  =  (4x*R)/T*,  or  reducing  R  to  feet  and  T  to  seconds 


GRAVITATION  :  !     1  0  1 


a  =  .00896.  This  value  of  a,  calculated  fiow,tli 
of  the  moon,  agrees  as  closely  with  the  preceding  value,  deduced 
from  the  law  of  gravitation,  as  could  be  expected  when  the  fact 
is  considered  that  only  approximate  values  for  the  various  con- 
stants have  been  used.  The  argument  must  be  considered  very 
strong  evidence  for  the  law  of  gravitation. 

145.  Force  of  Gravitation  Proportional  to  Mass.  —  According  to 
the  law  of  gravitation  the  attraction  between  two  bodies  is  pro- 
portional to  their  masses  and  is  independent  of  the  materials  of 
which  they  consist.  One  proof  of  this  was  given  by  Galileo, 
when  he  dropped  two  cannon  balls  of  different  sizes  from  the  lean- 
ing tower  of  Pisa  and  found  that  they  reached  the  ground  in  very 
nearly  the  same  time.  Their  accelerations  being  equal,  the  ratio 
of  the  force  to  the  mass,  must,  according  to  the  second  law  of 
motion,  be  the  same  for  both.  Yet  in  Galileo's  experiment  the 
larger  weight  was  slightly  ahead  of  the  smaller,  and  Galileo  cor- 
rectly explained  this  difference  by  remarking  that  the  air-friction 
would  be  proportionately  less  on  the  larger  body.  In  fact, 
because  of  this  air  friction  and  the  rapidity  of  the  motion,  it 
would  be  difficult  to  give  a  very  convincing  proof  of  the  law  by 
means  of  bodies  falling  with  the  full  acceleration  due  to  gravity. 

To  avoid  this  difficulty  Newton  experimented  with  a  pendulum, 
the  motion  of  which  depends  on  gravity  but  on  a  fraction  only  of 
the  full  force  of  gravity,  namely,  the  component  along  the  arc  of 
vibration.  The  bob  of  the  pendulum  was  a  thin  shell  and  into 
this  he  put  in  successive  experiments  different  substances.  In 
each  case  the  same  weight,  as  tested  by  weighing  with  a  balance, 
was  put  into  the  box  and,  since  the  force  of  air-friction  on  the 
box  for  the  same  amplitude  of  vibration  would  be  the  same  no 
matter  what  the  contents  of  the  box,  it  followed  that  at  a  given 
inclination  to  the  vertical  the  force  causing  the  motion  would  be 
always  the  same.  He  found  that  the  time  of  vibration  was  always 
the  same  no  matter  what  the  contents  of  the  box  and  hence  the 
masses  must  also  have  been  the  same;  that  is,  equal  masses  of 
different  substances  have  equal  weights.  These  experiments 
were  afterward  repeated  by  Bessel  with  much  greater  care  and 
with  the  same  result. 

The  above  experiments  prove  that  gravitation  is  not,  like  mag- 
netic attraction,  a  force  that  depends  on  some  quality  of  a  body 


102        MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

other  tii\aa;its  mass;  that  -is,  not  a  selective  force  but  a  general 
force.  That  it  does  not  depend  on  any  other  physical  condition 
such  as  temperature,  or  on  any  chemical  condition  such  as 
molecular  combination,  has  also  been  shown  by  most  careful 
weighing.  A  third  body  placed  between  two  bodies  has  not  the 
least  effect  in  shielding  them  from  their  mutual  attraction.  The 
fact  that  a  lump  of  gold,  when  hammered  out  into  an  exceedingly 
thin  sheet,  suffers  no  change  of  weight  shows  -that  the  weight 
of  a  body  does  not  depend  on  its  form,  that  gravity  acts  on  the 
particles  whether  surrounded  by  other  particles  of  the  same 
kind  or  not. 

146.  The  Constant  of  Gravitation.  —  The  law  of  gravitation  may 
be  stated  as  a  formula,  viz. 


where  G  is  a  constant  number  called  the  constant  of  gravitation. 
To  find  the  magnitude  of  G  it  is  necessary  to  measure  F  in  some 
case  where  m,  ra'  and  r  are  all  known.  This  was  first  done  by 
Henry  Cavendish  in  1797-8,  and  the  experiment,  usually 
called  the  Cavendish  experiment,  has  been  repeated  many  times 
since  with  increasing  care  and  accuracy.  Cavendish  suspended 
two  balls,  A  and  B,  from  the  ends  of  a  long  light  horizontal 

rod  which  was  supported  by  a  long  fine 
vertical  wire  attached  to  the  middle,  C, 
of  the  rod.  On  opposite  sides,  hori- 
zontally, of  the  balls  and  at  known  equal 
distances  he  placed  two  large  spheres  of 
lead,  P  and  Q.  The  attraction  between 
each  ball  and  the  adjacent  large  sphere 
had  a  moment  about  C  that  produced  a 
twist  of  the  supporting  wire.  When 
FIO.  78-—  Principle  of  the  the  spheres  were  in  the  position  PlQl, 

Cavendish  experiment.         .,  .   ,  •  -,'•  i 

the  twist  was  in  one  direction,  and  when 

they  were  in  the  position  P2Q2  the  twist  was  in  the  opposite 
direction.  To  deduce  the  force  of  attraction  from  the  magni- 
tude of  the  twist,  the  constant  of  torsion  of  the  wire  (§119)  had 
to  be  found  by  timing  vibrations  of  the  wire,  when  the  spheres 
were  removed  to  positions  where  they  had  no  influence  on  the 


GRAVITATION  103 

vibrations  of  AB.  Thus  Ft  m,  m',  and  r  were  found  and  when 
they  were  substituted  in  the  above  formula  the  value  of  G  was 
obtained. 

In  more  recent  work  the  apparatus  has  been  greatly  improved.  The 
greatest  improvement  has  been  in  the  substitution  of  very  fine  quartz  thread 
for  the  wire.  This  also  permitted  of  the  apparatus  being  greatly  reduced 
in  size,  so  that,  whereas  AB  in  Cavendish's  experiment  was  6  ft.  long,  in 
Boys'  apparatus  it  was  only  0.9  inch,  and  the  masses  A,  B,  andP,  Q,  were 
also  greatly  reduced  in  size.  The  value  obtained  for  G  (using  c.g.s.  units) 
was  6.6579  X10~8;  this  is,  therefore,  the  force  in  dynes  of  the  attraction 
between  spheres  of  one  gram  each  at  a  distance  of  1  cm.  between  their 
centers.  When  it  is  remembered  that  a  dyne  is  about  the  weight  of  a  milli- 
gram, it  is  seen  that  the  force  measured  in  the  above  experiments  must 
be  exceedingly  small;  hence  the  difficulty  of  the  experiment. 

147.  The  Mean  Density  of  the  Earth. — The  determination  of  G 
made  it  possible  to  calculate  the  mass  of  the  earth  (hence  Caven- 
dish is  sometimes  said  to  have  been  the  first  to  "  weigh  the  earth  ") . 
For  if  m'  in  the  formula  for  the  law  of  gravitation  be  put  equal 
to  one  gram  and  m  and  r  be  taken  as  respectively  the  mass  and 
radius  of  the  earth,  F  will  be  the  force  of  attraction  between  the 
earth  and  a  body  of  1  gm.  mass  and  this  is,  as  we  know,  980  dynes. 
Thus  the  formula  gives  us  the  value  of  m,  the  mass  of  the  earth, 
which  is  found  to  be  5.97  X 1027  gms.  This  figure  is  so  large  as 
to  convey  no  distinct  meaning,  but  a  different  way  of  stating  the 
result  will  be  more  easily  comprehended.  The  density  of  a  homo- 
geneous body  such  as  water  is  its  mass  per  unit  volume,  and 
when  the  density  of  a  body  is  not  everywhere  the  same  we  may 
speak  of  its  mean  density  or  its  whole  mass  divided  by  its  whole 
volume.  Thus  to  get  the  mean  density  of  the  earth  we  divide 
its  whole  mass,  as  given  above,  by  its  whole  volume.  The  result 
is  5.527,  that  is  to  say,  on  the  average  the  earth  is  5.527  times  as 
dense  as  water.  It  is  remarkable  that  Newton,  reasoning  from 
the  very  slight  evidence  available  in  his  time,  supposed  the  mean 
density  of  the  earth  to  be  between  5  and  6. 

148.  The  Tides. — At  any  place  on  the  shore  of  an  ocean  the 
level  of  the  water  rises  to  a  maximum  and  falls  to  a  minimum 
once  in  about  every  twelve  hours  and  25  minutes.  These  risings 
and  fallings  are  called  the  tides.  They  are  due  to  the  forces  of 
attraction  which  the  moon  and  the  sun  exercise  on  the  water  on 


104        MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

the  surface  of  the  earth  and  to  the  rotation  of  the  earth.  The 
complete  explanation  of  their  action  is  extremely  difficult,  owing 
to  the  irregularities  of  the  continents  and  to  other  causes. 

UNITS 

149.  Fundamental  and  Derived  Units. — The  measurement  of  any 
quantity  consists  in  comparing  it  with  a  unit  of  the  same  kind 
(§2).     Thus  a  length  is  measured  by  comparing  it  with  a  unit 
of  length,  such  as  the  foot  or  meter;  a  velocity  is  measured  by 
comparing  it  with  a  unit  of  velocity,  such  as  a  foot  per  second 
and  so  on.     Hence  we  need  as  many  units  as  there  are  different 
kinds  of  quantities  to  be  measured. 

But  all  these  necessary  units  are  not  necessarily  independent. 
It  is  found  that  in  Mechanics  three  independent  or  fundamental 
units  are  sufficient;  all  others  can  be  defined  in  terms  of  these.  A 
unit  defined  by  reference  to  some  other  unit  or  units  is  called  a 
derived  unit. 

150.  Absolute  Systems  of  Units. — A  system  of  units  in  which 
the  derived  units  bear  the  simplest  possible  relation  to  the  funda- 
mental units  is  called  an  absolute  system.     In  such  a  system  the 
unit  of  area  or  surface  is  the  square  of  the  unit  of  length,  the  unit 
of  volume  is  the  cube  of  the  unit  of  length,  the  unit  of  velocity 
is  a  velocity  of  unit  length  per  unit  time,  and  so  on.     Given  any 
three  fundamental  units  of  length,  time  and  mass,  we  can  build 
up  an  absolute  system  of  derived  units.     Thus  we  have  one 
absolute  system  founded  on  the  cm.,  gm.,  and  sec.,  another 
founded  on  the  ft.,  lb.,  and  sec.,  and  so  on. 

151.  Dimensions  of  Units. — It  is  sometimes  necessary  to  translate  re- 
sults from  one  absolute  system  to  another.  It  then  becomes  necessary  to 
consider  how  the  magnitude  of  a  derived  unit  changes  when  the  funda- 
mental units  are  changed.  For  this  purpose  we  need  to  know  the  dimen- 
sions of  the  derived  unit,  that  is,  the  powers  of  the  fundamental  units  to 
which  the  derived  unit  is  proportional.  For  instance,  the  unit  of  area  is 
the  square  of  the  unit  of  length,  or  area  is  of  2  dimensions  in  length,  a 
statement  briefly  summarized  by  the  dimensional  formula  [<A]=[L]2; 
similarly,  using  [Vol]  for  the  unit  of  volume,  [Vol]=[L]*. 

1555.  Dimensions  of  Velocity. — The  unit  of  velocity  is  denned  in  terms 
of  the  unit  of  length  and  the  unit  of  time.  To  find  the  dimensions  in 
these  units  consider  any  relation  between  velocity,  length,  and  time,  such 
as  s-*vt  (§19).  This  is  a  relation  between  numerical  measures  (§2), 
but  it  implies  certain  relations  between  the  units  used  in  measuring  these 


UNITS  105 

quantities;  both  sides  must  be  of  the  same  dimensions  in  fundamental 
units,  or  they  could  not  be  equal.  Hence,  if  we  denote  the  unit  of  velocity 
by  [7],  [L]=[VH[T]  or  [7]-[LJT]-V  Thus  velocity  is  of  +1  dimension 
in  length  and  —  1  dimension  in  time. 

163.  Dimensions  of  Acceleration. — Consider  any  relation  between   ac- 
celeration length  and  time,  such  as  s=-$crf2.    From  this  by  the  line  of 
reasoning  explained  in  the  last  section  we  derive  at  once  [7j]*»[AJT]*. 
Hence  [A]=*[L][T]-*.    The  sign   of   equality  in  such  expressions  denotes 
equality  of  dimensions.     Constant  numerical  factors  (such  as  the  i  above) 
are  of  zero  dimensions,  that  is,  they  do  not  change  when  we  change  the 
fundamental  units. 

164.  Other  Derived  Units. — The  above  examples  sufficiently  explain  the 
method  by  which  the  following  table  is  derived. 

TABLE  OF  DERIVED  UNITS  USED  IN  MECHANICS 

Name  of  Unit 

Relation  of  Dimensions          in  c.g.s. 

Quantity.                           Numerics.  of  Units.            System. 

Linear  velocity,  v                              s  =  vt  [L][T]-1 

Linear  acceleration,  a                        s=*$at*  [L][T]~* 

Angular  velocity,  at                           <f>=a)t  [T]~l 

Angular  acceleration,  a                    0=»icrf2  |T]-2 

Force,  F                                            F  =  ma  [L]  [T]  -*[M]            dyne 

Moment  of  Force,  L                          L  =  Fp  [L]*[T]  ~2[M] 

Moment  of  Inertia,  7                          7  =  mr2  [L]*[M] 

Work,  W                                          W=Fs  [L]*[T]-2[M]          erg 

Kinetic  Energy,  E.                      K.E.  —  %mv*  [L]2[T]  -2[M ]           erg 

Potential  Energy                          P.E.^Fs  [L]*[T]-*[M]          erg 

155.  Examples  of  Use  of  Dimensional  Relations. — Where  a  derived 
unit  has  no  particular  name  its  dimensional  formula  is  a  sufficient  name. 
Thus  the  unit  of  acceleration  has  no  special  name  and  10  units  of  accelera- 
tion in  the  C.G.S.  system  is  written 

cm. 
secT" 

A  frequent  use  of  dimensional  relations  is  in  changing  the  measure  of  a 
quantity  from  one  absolute  system  of  units  to  another.  For  example,  the 

acceleration  of  gravity  is  980 -.  what  is  it  in  — : —  ?     Suppose  it  is  x. 

sec.2  min.» 

Then 

meter  cm.  cm.  /minAa  * 

x -  =  980— .'.3  =  980 I-   -I  -35,280. 

mm. 2  sec.2  m.    y  sec.  / 

Another  use  of  these  relations  is  in  testing  the  accuracy  of  complicated 
formulas.  The  two  sides  of  the  equation  must  be  of  the  same  dimensions 
or  they  could  not  stand  for  the  same  thing. 


106        MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

PROPERTIES  OF  MATTER 
Constitution  of  Matter 

166.  In  the  preceding  chapters  on  the  principles  of  Mechanics, 
we  have  had  (with  slight  exceptions)  to  consider  matter  from  but 
one  point  of  view,  namely,  its  inertia.     The  forces  that  the  par- 
ticles of  a  body  exert  on  one  another  did  not  need  to  be  considered, 
for  they  cancelled  out  when  the  action  of  the  body  as  a  whole 
was  considered. 

We  shall  now  consider  other  important  properties  of  matter, 
especially  those  which  depend  on  the  force  between  particles.  It 
will  be  seen  that  the  connections  between  these  properties  are  not 
so  well  understood  as  the  relations  between  the  quantities  studied 
in  mechanics.  This  is  chiefly  because  the  ultimate  particles  of  a 
body  are  so  small  that  they  cannot  be  studied  separately.  In  fact 
we  can  only  infer  their  existence  and  relations  from  the  proper- 
ties they  exhibit  in  the  large  groups  which  we  call  bodies. 

167.  The  Three  States  of  Matter. — Following  popular  language 
we  classify  bodies  as  solids  and  fluids.     The  characteristic  of  a 
solid  is  that  it  has  a  definite  shape  which  it  does  not  readily  relin- 
quish, while  a  fluid  flows  easily  or  changes  its  shape  in  response  to 
the  smallest  influence.     (It  will  be  seen  later  that  the  distinction 
is  not  quite  definite,  that  some  bodies  lie  on  the  borderland 
between  the  two  classes.)      The  particles  of  a  solid  are  held  in 
(practically)   fixed  positions  by  the  forces  between  them,  but 
each  particle  has  a  freedom  to  vibrate  about  its  mean  position 
(see  §161). 

Fluids  are  divided  into  liquids  and  gases.  The  peculiarity  of  a 
liquid  is  that,  while  it  readily  flows,  it  has  a  definite  volume  which 
it  does  not  readily  change.  A  gas  yields  to  the  smallest  force  ex- 
erted to  change  its  volume,  in  other  words,  it  has  no  definite 
volume  of  its  own,  but  takes  the  volume  of  the  containing  vessel 
however  large.  (This  distinction  also  is  only  general.)  The 
particles  of  a  liquid  are  close  together  and  attract  each  other 
with  powerful  forces.  These  forces  react  strongly  against  outside 
forces  that  tend  to  change  the  mean  distance  between  the  parti- 
cles, but  they  are  such  as  to  permit  sliding  motions  of  the  particles. 
The  particles  of  a  gas  are  practically  separate  bodies  flying  in 
space  and  exerting  no  appreciable  forces  on  one  another  except 
at  impact  of  particle  on  particle. 


PROPERTIES  OF  MATTER  107 

168.  Elements  and  Compounds. — In  innumerable  cases  two  or 
more  substances  coalesce  to  form  a  new  substance  that  may  be 
so  distinct  in  all  its  properties  that  nothing  apparently  remains 
to  suggest  the  constituents  from  which  it  was  formed.     Thus 
two  substances,  oxygen  and  hydrogen,  gaseous  under  ordinary 
conditions,  combine  to  form  a  liquid,  water.   Harmless  substances 
may  on  combination  form  deadly  poisons  or  explosives.     Sub- 
stances that  may  be  made  from  constituents  which  have  proper- 
ties distinct  from  the  resultant  are  called  compounds. 

Conversely,  compounds  may  be  divided  up  into  constituents 
differing  widely  from  the  original  substance  and  these  constituents 
may  be  themselves  capable  of  being  resolved  into  other  constitu- 
ents. But  there  are  many  substances  which  have  not  as  yet  been 
resolved  into  constituents  and  such  are  called  elements.  Of  these 
there  are  about  80  known. 

169.  Molecules  and  Atoms. — Many  facts,  chiefly  such  as  are 
more  closely  studied  in  chemistry,  justify  the  belief  that  (1)  an 
element  consists  of  very  small  particles  called  atoms,  (2)  all  the 
atoms  in  one  elementary  body  are  identical  in  size  and  other  prop- 
erties, but  different  from  those  of  any  other  elementary  body, 
(3)  these  atoms  are  combined  in  similar  groups  called  molecules 
(in  some  substances  the  atom  and  the  molecule  are  identical). 
It  is  also  believed  that  a  compound  consists  of  molecules  and 
that  each  molecule  consists  of  two  or  more  atoms  of  the  constitu- 
ents of  the  compound.     There  is  also  much  reason  to  believe  that 
in  many  substances,  especially  liquids  and  solids,  molecules  are 
frequently  combined  to  form  groups  or  molecular  aggregates  of 
two  or  more  molecules  each. 

Molecules  and  atoms  are  extremely  small  and  will  probably 
never  be  separately  visible,  however  much  optical  instruments 
may  be  improved.  Thus  in  a  cubic  centimeter  of  a  gas  under 
ordinary  conditions  there  are  about  4X1019  molecules. 

There  is  good  reason  to  believe  that  atoms  contain  still  smaller 
parts  called  electrons,  which  may  pass  from  atom  to  atom  and 
are  sometimes  entirely  separated  from  atoms.  The  properties 
of  these  explain  many  of  the  phenomena  of  light  and  electricity. 

160.  Interniolecular  Forces. — It  is  evident  from  the  great 
forces  necessary  to  pull  a  solid  body  apart  that  there  are  com- 
paratively great  forces  between  particles;  but  the  ease  with 


108        MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

which  a  brittle  body  falls  apart  when  a  slight  crack  appears  shows 
that  the  forces  are  only  appreciable  when  the  attracting  particles 
are  very  close  together.  The  latter  point  is  also  shown  by  the 
fact  that  a  body  reduced  to  powder,  e.g.,  the  graphite  of  which 
lead  pencils  are  made,  can  only  be  changed  back  into  a  com- 
pact solid  by  intense  pressure. 

Roughly  speaking,  it  may  be  said  that  the  force  of  molecular 
attraction  in  water  is  inappreciable  at  distances  greater  than 
about  .00000015  cm.  The  magnitude  and  the  range  of  the  inter- 
molecular  forces  are,  of  course,  different  for  different  substances, 
and  the  characteristic  properties  of  different  substances  probably 
depend  on  these  differences. 

161.  Kinetic  Theory  of  Matter. — There  is  very  strong  evidence 
that  the  particles  of  which  bodies  are  made  up  are  in  no  case  at 
rest.     Thus  two  different  gases  contained  in  two  different  vessels 
mix  with  great  rapidity  when  the  vessels  are  put  in  communication. 
This  process  is  called  diffusion.     Liquids  will  also  diffuse  into  one 
another  (except  non-miscible  liquids  like  oil  and  water) ,  though 
much  more  slowly  than  gases,  because  of  the  greater  closeness  of 
the  particles  and  the  frequent  changes  of  direction  of  motion  of 
a  particle  produced  by  impact  on  other  particles.     Even  many 
solids  show  by  diffusion  that  their  particles  are  not  at  rest;  thus 
when  a  small  block  of  gold  is  placed  on  a  block  of  lead  with  planed 
surfaces  in  close  contact,  after  the  lapse  of  some  weeks  it  is  possible 
to  detect  particles  of  gold  which  have  wandered  into  the  lead  and 
vice  versa.     There  are  many  other  reasons  for  believing  that  the 
particles  of  matter  are  in  all  cases  in  motion.     This  hypothesis 
is  called  the  hypothesis  of  the  kinetic  constitution  of  matter. 

162.  Density  and  Specific  Gravity. — The  density  of  a  body  is  its 
mass  per  unit  volume.     If  the  masses  of  all  equal  volumes  of  the 
body  are  the  same,  the  density  is  uniform  and  equal  to  the  mass  in 
any  unit  of  volume.     If  the  masses  of  equal  volumes  are  not  the 
same,  the  density  is  not  uniform.     The  mean  density  in  any  par- 
ticular volume  of  the  body  is  the  mass  in  that  volume  divided  by 
the  volume.     The  density  at  a  point  is  the  mean  density  in  a  small 
volume  enclosing  the  point  when  the  volume  is  supposed  to  be 
decreased  without  limit. 

The  measure  of  the  density  of  a  body  depends,  of  course,  on  the 
units  of  mass  and  volume  employed.  If  the  c.g.s.  system  is 


PROPERTIES  OF  SOLIDS  109 

employed,  density  is  the  number  of  gms.  per  c.c.  In  this  system 
the  density  of  water  at  4°C.  is  very  nearly  unity,  since  the  gram 
was  originally  intended  to  be  the  mass  of  1  c.c.  of  water  at  4°C. 
In  the  British  system  the  density  of  a  body  is  the  number  of  Ibs. 
per  cu.  ft.  of  a  body.  In  this  system  the  density  of  water  is 
62.4,  since  that  is  the  number  of  Ibs.  in  a  cu.  ft.  of  water. 

The  specific  gravity  of  a  body  is  the  ratio  of  its  density  to  that 
of  some  standard  substance.  The  standard  usually  employed  is 
water  at  4°G.  Thus  if  D  be  the  density  of  a  body  and  d  that  of 
water  at  4°G.  the  specific  gravity  of  the  body  is  D/d.  Now  in 
the  c.g.s.  system  d  is  very  nearly  unity.  Hence  in  this  system 
density  and  specific  gravity  are  numerically  equal.  But  in  the 
British  system,  since  d  is  62.4,  the  specific  gravity  of  a  body  is 
its  density  divided  by  62.4. 

TABLE  OP  DENSITIES  (GMS.  PER  CM.*) 

Aluminium 2.60  Iron  (about) 7.60 

Brass  (about) -  . .     8.50  Lead 11.37 

Copper 8.92  Platinum 21 .50 

Gold 19.32  Silver 10.53 

Ice 917  Air  at  0°  and  1  atmo...      .001293 

Alcohol  at  20° 789  H    "  "      "    "" 00008988 

Ether     "    " 715  N    ""      "    "" 001256 

Mercury"   " 13.55  O     ""      "    "" 001430 

Sea  Water"..  1.026 


PROPERTIES  OF  SOLIDS 

163.  Homogeneity  and  Isotropy. — A  homogeneous  body  is  one 
which  has  at  all  points  the  same  properties,  so  that  small  spheres 
of  equal  radii  cut  out  of  different  parts  of  the  body  would  be 
identical  in  properties.  Many  crystals  are  nearly  perfectly  homo- 
geneous, and  so,  too,  is  good  glass,  such  as  plate  glass  or  the  glass 
of  lenses.  Many  other  bodies  are  approximately  homogeneous, 
for  example,  most  metals,  wood,  stones,  etc. 

An  isotropic  body  is  one  which  has  at  any  point  the  same  prop- 
erties in  all  directions,  so  that  if  at  any  point  a  sphere  were  cut 
out  there  would  be  nothing  in  the  properties  of  the  sphere  to 
indicate  the  original  direction  of  any  diameter.  All  liquids  and 


110        MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

gases  are  isotropic  under  ordinary  conditions  but  many  sub- 
stances, such  as  crystals,  woods  and  drawn  metals,  are  distinctly 
non-isotropic. 

164.  Elasticity. — When  the  shape  or  volume  of  a  solid  is  changed 
by  the  application  of  some  force,  there  is  in  most  cases  a  tendency 
to  return  to  the  original  shape  or  volume  when  this  force  is  re- 
moved. This  tendency  to  recover  from  distortion  is  called  elas- 
ticity. It  is  one  of  the  most  important  properties  of  a  solid,  since 
the  usefulness  of  many  bodies  such  as  springs,  musical  instru- 
ments, etc.,  depends  on  the  extent  to  which  they  possess  this 
property.  It  is,  therefore,  a  property  that  has  been  very  exten- 
sively studied. 

166.  Strain. — Any  change  of  shape  or  of  volume  or  of  both  is 
called  a  strain.  Thus  the  bending  of  a  beam,  the  twisting  of  a 
rod,  the  compression  of  a  liquid  or  a  gas  into  a  smaller  volume  are 
strains.  The  term  strain  is  a  geometrical  one  and  its  definition 
contains  no  reference  to  force  or  energy,  although,  as  we  shall 
see,  force  and  energy  are  present  when  a  body  is  in  a  state  of 
strain. 

A  strain  that  consists  in  a  change  of  shape  only  without  any 
change  of  volume  is  called  a  shear.  The  strain  of  a  moderately 
twisted  wire  or  rod  is  a  shear. 

A  strain  that  consists  in  a  change  of  volume  only  without  any 
change  of  shape  has  not  received  any  special  name,  but  we  may 
for  brevity  call  it  a  volume-strain.  Such,  for  example,  is  the 
strain  of  a  sphere  of  cork  or  of  any  isotropic  body  when  placed 
in  a  fluid  which  is  subjected  to  great  pressure  in  a  closed 
vessel. 

While  for  simplicity  we  have  first  enumerated  strains  in  which 
either  volume  or  shape  alone  changes,  strains  which  involve 
changes  of  both  are  more  common.  Thus  the  stretching  of  a  wire, 
the  compression  of  a  pillar,  the  bending  of  a  beam,  etc.,  are  strains 
of  both  volume  and  shape.  A  body  is  said  to  be  homogeneously 
strained,  or  the  strain  is  described  as  homogeneous,  when  the  na- 
ture and  magnitude  of  the  strain  is  the  same  at  all  points  in  the 
body.  Thus,  when  a  wire  is  stretched  or  a  rod  compressed  and 
when  a  liquid  or  gas  is  subjected  to  pressure,  the  strain  is  homo- 
geneous. But,  when  a  wire  or  rod  is  twisted,  the  strain  is  great- 
est at  the  surface  and  least  at  the  center,  and.  when  a  beam  is 


PROPERTIES  OF  SOLIDS  111 

bent,  there  is  a  stretching  on  the  convex  side  and  a  compression 
on  the  concave  side  and  the  strain  is  heterogeneous. 

166.  Stress. — When  a  body  is  in  a  state  of  strain  owing  to  the 
action  of  external  forces  on  it,  there  are  internal  forces  between 
contiguous  parts  of  the  body  in  addition  to  whatever  internal 
forces  there  may  have  been  before  the  strain  occurred.  If  at  a 
point  a  dividing  plane  be  imagined,  the  part  of  the  body  on  one 
side  will  act  with  a  certain  force  on  the  part  on  the  other  side 
and  the  latter  will  react  with  an  equal  and  opposite  force.  These 
two  forces  together,  the  action  and  the  reaction,  constitute  a  stress. 
In  some  cases,  as  we  shall  see,  the  stress  is  perpendicular  to  the 
imaginary  dividing  plane  and  in  others  parallel  to  it,  but  in  any 
case  the  magnitude  of  the  stress  is  the  force  per  unit  area  of  such 
an  imaginary  dividing  plane. 

The  terms  homogeneous  and  heterogeneous  apply  to  stress  just 
as  to  strain.  In  many  cases,  for  example  in  the  stretch  of  a  wire 
by  an  attached  weight,  the  stress  in  a  body  is  equal  to  the  ex- 
ternal force  per  unit  area  that  acts  on  the  body  and  produces  the 
strain,  and  in  such  cases  we  may  speak  of  this  external  force  per 
unit  of  area  as  the  stress.  In  other  cases,  as,  for  example,  in 
the  bending  of  a  beam  by  a  weight  acting  at  some  point,  the 
stress  does  not  bear  a  simple  relation  to  the  external  force  and  we 
must  take  care  to  distinguish  them. 

167.  The  Measure  of  a  Strain  and  of  a  Stress. — A  strain  which 
consists  in  a  change  of  volume  only  is  measured  by  the  proportion 
in  which  the  volume  is  changed.  If  the  strain  is  homogeneous  the 
measure  may  be  taken  as  the  change  in  unit  volume,  or  if  a  volume 
V.  becomes  (V+v)  the  measure  of  the  strain  is  v/V.  If  the 
strain  is  not  homogeneous  the  measure  of  the  strain  at  any  par- 
ticular point  is  the  value  of  v/  V  at  the  point,  when  V  is  taken  as 
the  volume  of  an  indefinitely  small  portion  surrounding  the  point. 
To  produce  this  change  of  volume  force  must  be  applied  to  the 
surface  of  the  solid  in  the  form  of  either  pressure  or  a  ten- 
sion, and  inside  the  body  each  part  will  press  or  pull  on  each 
neighboring  part.  The  amount  of  this  pressure  or  pull  per  unit 
area  is  the  measure  of  the  stress. 

The  measure  of  a  shear  will  be  most  readily  understood  by  con- 
sidering the  simplest  way  in  which  a  shear  may  be  produced. 
Consider,  for  example,  a  rectangular  block  of  a  firm  jelly  between 


112        MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

two  boards  to  which  it  adheres.     Let  PQRS  be  one  rectangular 
face  and  PQ,  RS  the  edges  of  the  boards.     Apply  to  the  boards 
equal  and  opposite  forces  parallel  to  them  and  to  the  face 
PQRS.      The  face  PQRS   is   changed   to  the  form   P^RS. 
Each  section  of  the  block  parallel  to  the  boards  moves  parallel 
to  itself  a  distance  proportional  to  its  distance 
from  RS.     Each  of  the  right  angles  of  PQRS  is 
changed  by  the  same  amount,  say  6,  and  this 
change  is  the  measure  of  the  shear.     When  6  is 
small,  as  it  is  in  most  practical  cases,  the  mag- 
nitude of  the  angle  0  in  radian  measurement  is 
FIO.  7».-shear  and   P^P^PS,    or   taking   PS   equal   to  unity,  the 
•hearing  stress.       relative  displacement  of  two  planes  unit  distance 

apart. 

If  an  imaginary  plane  be  supposed  drawn  anywhere  in  the 
block  parallel  to  the  boards,  the  part  on  one  side  of  this  plane  will 
exert  a  tangential  force  on  the  part  on  the  other  side  and  this 
force  will  equal  the  force  applied  to  the  boards.  The  magnitude 
of  the  force  per  unit  area  is  the  measure  of  the  shearing  stress. 

While  we  have  referred  only  to  the  forces  parallel  to  PQ  and  RS,  it  is 
clear  that  the  shear  cannot  be  produced  without  other  forces  applied  to  the 
block.  If  only  the  two  forces  described  were  applied,  the  block  would  not 
be  at  rest  but  in  rotation,  since  the  two  constitute  a  couple.  The  effect  is 
readily  perceived  when  the  attempt  is  made  to  apply  the  two  opposite  forces. 
It  is,  in  fact,  necessary  to  also  apply  other  forces  forming  an  opposite 
counterbalancing  couple,  say  along  SPl  and  QVR.  The  effect  of  all  four 
forces  is  to  produce  a  stretch  along  RPl  and  a  compression  along  QVS  and 
the  proportional  stretch  is  equal  to  the  proportional  compression,  since 
there  is  no  change  of  volume. 

168.  Hooke's  Law. — When  any  body  is  strained  beyond  a  cer- 
tain amount  and  then  released,  it  fails  to  return  completely  to  its 
original  form  and  volume  or  it  retains  a  permanent  set.  The 
largest  strain  of  any  kind  which  a  body  may  undergo  and  still 
completely  recover  from  when  released  is  called  the  limit  of  elas- 
ticity for  that  form  of  strain,  and  the  corresponding  stress  is 
called  the  limiting  stress.  The  limit  of  elasticity  is,  of  course, 
widely  different  for  different  substances.  Thus,  rubber  may  be 
greatly  extended  and  yet  recover,  while  the  limit  for  glass  and 


PROPERTIES  OF  SOLIDS  113 

ivory  is  very  small.  (Cases  in  which  the  limit  is  somewhat 
indefinite  will  be  considered  later.) 

Within  the  limit  of  elasticity  a  simple  law,  first  stated  by  Hooke 
in  1676  and  known  as  Hooke' s  law,  holds,  namely,  "stress  is  pro- 
portional to  strain."  (Hooke's  statement  in  Latin  was  "Ut  tensio 
sic  vis. ")  Hooke  illustrated  his  law  by  various  cases  of  strain, 
such  as  the  stretching  of  a  spiral  spring  and  of  a  wire,  the  bending 
of  a  beam,  the  twisting  of  a  wire  and  so  on. 

169.  Moduli  of  Elasticity. — While  elasticity  has  already  been  de- 
fined as  the  tendency  of  a  body  to  recover  its  shape  or  volume 
when  distorted,  the  definition  is  purely  qualitative  and  affords 
no  means  of  assigning  a  numerical  value  to  the  elasticity  of  a 
substance.  A  quantitative  definition  of  the  elasticity  of  a  sub- 
stance for  any  form  of  strain  follows  from  Hooke's  law.  The 
measure  or  modulus  of  elasticity  is  the  ratio  of  the  magnitude  of 
the  stress  to  that  of  the  accompanying  strain,  this  ratio  being  a 
constant  within  the  limits  of  elasticity.  As  there  is  a  great  variety 
of  forms  of  strain  there  is  a  correspondingly  large  number  of 
moduli  of  elasticity  for  any  substance;  but  only  a  few  of  these 
are  important  enough  to  be  enumerated. 

When  the  strain  is  one  of  volume  only  the  elasticity  is  called 
elasticity  of  volume.  The  modulus  of  elasticity  of  volume  or  the 
bulk-modulus,  as  it  is  frequently  called,  is  the  ratio  of  the  stress, 
or  the  pressure  per  unit  area,  P,  to  the  change  of  volume  per 
unit  volume.  The  bulk  modulus  of  a  substance  is  usually  denoted 
by  k.  Hence,  if  a  volume  V  undergoes  a  change  of  volume  v  and 
the  stress  is  P,  * 


V          V 

~V 

The  reciprocal  of  the  bulk  modulus  is  called  the  coefficient  of 
compressibility  of  the  substance.  It  means  the  ratio  of  the  pro- 
portional compression  to  the  pressure  per  unit  area,  or,  supposing, 
the  latter  to  be  unity,  the  coefficient  of  compressibility  is  the  ratio 
in  which  the  volume  is  reduced  by  unit  pressure  per  unit  area. 

When  the  strain  is  a  shear  the  modulus  of  elasticity  is  called 
the  shear  modulus,  or  often  the  simple  rigidity,  and  is  the  ratio 
of  the  shearing  stress  to  the  shear.  Denoting  the  shearing  stress 


114        MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

by  T,  the  shear  corresponding  to  T  by  0,  and  the  shear  modulus 
byn, 

T 


170.  Torsion.  —  When  a  wire  or  rod  of  homogeneous  isotropic  material 
is  twisted,  we  may  imagine  the  whole  length  divided  into  transverse  slices 

of  equal  thickness  by  planes  perpendicular  to  the  axis. 
Each  such  slice  will  be  rotated  about  the  axis  to  an  extent 
proportional  to  its  distance  from  the  fixed  end.  Moreover, 
one  face  of  each  slice  (the  one  farthest  from  the  fixed 
end)  will  be  rotated  more  than  the  other.  Let  us  now 
suppose  that  each  slice  is  very  thin,  and  that  it  is  divided 
up  before  twisting  into  very  small  cubes  (or  nearly  cubes) 
by  a  series  of  imaginary  planes  through  the  axis  inter- 
HIII  I  I  H  I  IIHII  sected  by  concentric  cylinders.  Thus  each  cube  will  have 
(  "'  four  edges  parallel  to  the  axis,  four  others  in  the  direction 

Y/////  f7  /  ////A      of  radii,  while  the  remaining  four  will  be  short  and  practi- 
(c)  cally  straight  arcs  of  circles.     After  the  twist  each  cube 

Fio.  80.—  Shear  w^  have  a  strain  like  the  cube  of  jelly  in  §167.  Hence 
of  a  small  cube  in  the  strain  is  a  shear,  but,  since  the  strain  of  each  cube  will 
a  twisted  wire.  De  proportional  to  its  distance  from  the  axis,  the  strain  is 

not  homogeneous. 
The  constant  of  torsion  of  a  wire  has  already  been  defined  in  §119. 

171.  Young's  Modulus.  —  A  very  frequent  form  of  strain  is  that 
of  a  uniform  wire  or  rod  which  is  clamped  at  one  end  and  is 
acted  on  by  a  longitudinal  force  at  the  other  end.     Such  a  strain 
is  called  a  stretch.     Any  short  part  of  the  wire  is  extended  in 
the  same  proportion  as  the  whole  wire.     The  measure  of  the 
stretch  is  the  extension  per  unit  length,  or,  denoting  the  un- 
stretched  length  of  the  wire  by  L  and  the  total  extension  by  I,  the 
stretch  is  l/L.     The  measure  of  the  stress  is  the  external  pull 
per  unit  of  cross-sectional  area.     Denoting  by  F  the  whole  force 
applied  to  one  end  and  by  a  the  cross-sectional  area  of  the  wire, 
the  pull  per  unit  area  anywhere  in  the  rod  due  to  the  force  F  is 
F/a,  which  is,  therefore,  the  measure  of  the  stress.     Young's 
modulus,  which  we  may  denote  by  M  ,  is,  therefore,  (F/a)  -*-  (l/L) 
or 

M    FL 

M  =  —  r 

al 

For  some  common  materials  the  average  values  of  k,  n,  and  M  in 
c.g.s.  units,  that  is,  dynes  per  cm2  are  as  follows: 


PROPERTIES  OF  SOLIDS 


115 


k. 

Copper 17X1011 

Glass 4X10" 

Iron  (wrought) 15X10" 

Lead 4X10" 

Steel..  17X10" 


n. 

4X10" 
2X10" 
7X10" 
2X10" 
8X10" 


M. 
11X10" 

6X10" 
19X10" 

IX 10" 
23X10" 


172.  Volume  Changes  when  a  Wire  is  Stretched. — When  a  wire  or  rod 
is  stretched,  there  is  obviously  a  change  of  shape  in  every  part  of  the  wire 
or  rod,  for  the  length  is  increased  while  the  cross-section  is  decreased. 
Whether  a  change  of  volume  also  occurs  can  only  be  determined  by  experi- 
ment. If  the  cross-section  diminishes  in  the  same  proportion  as  that 
in  which  the  length  increases,  there  is  no  change  of  volume;  whereas,  if 
the  proportion  in  which  the  length  increases  exceeds  that  in  which  the 
cross-section  diminishes,  there  is  an  increase  of  volume.  Careful  experi- 
ment shows  that  in  all  cases  there  is  an  increase  of  volume;  but  in  some 
substances,  e.g.,  india  rubber,  the  change  of  volume  is  very  small. 


B 


N 


-N 


173.  Flexure. — A  very  common  strain  closely  related  to  stretch- 
ing is  that  of  a  plank  supported  at  both  ends  and  carrying  a  load 
at  the  middle,  or  supported  at  the  middle  and  loaded  at  each  end, 
or  clamped  horizontally  at  one  end  and  loaded  at  the  other  end. 
A  little  consideration  will  make  it  clear 
that  in  these  cases  we  have  to  do  with 
stretches  and  shortenings  such  as  those 
already  discussed.  If  we  suppose  the 
plank  divided  into  a  large  number  of 
longitudinal  strips,  the  strips  on  the  con- 
vex side  are  stretched  by  the  bending, 
while  those  on  the  concave  side  are 
shortened.  There  must,  of  course,  be 
an  intermediate  surface  where  there  is 
neither  stretch  nor  compression  and  this 
surface  is  called  the  neutral  surface.  The 
extension  or  compression  of  any  strip  is  proportional  to  its 
distance  from  the  neutral  surface.  Thus  the  strain,  while  not 
homogeneous,  is  everywhere  of  the  nature  of  an  extension  or  a 
compression  and  Young's  modulus  is  the  only  modulus  involved. 
If  a  bar  of  length  I,  breadth  6,  and  depth  d  be  supported  at  both 
ends  and  be  subjected  to  a  perpendicular  force  F  at  the  middle, 
the  depression  produced  is  Fl*/4:Mbds. 


Fia.  81. — Bending  of  a  beam 
(exaggerated). 


116        MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

174.  Direct  Impact  of  Elastic  Bodies. — When  two  bodies  in 
motion  collide,  each  exerts  a  momentary  force  on  the  other  and 
each,  therefore,  suffers  a  change  of  velocity.  The  result  is  diffi- 
cult to  calculate  except  in  certain  simple  cases. 

When  the  bodies  are  uniform  spheres  and  are  moving  before 
impact  along  the  line  joining  their  centers,  the  result  can  be 
calculated.  Let  the  masses  be  m  and  m'  and  the  velocities 
before  impact  u  and  u' ',  and  suppose  that  both 
are  in  the  positive  direction  and  u  >  u'.  After 
the  impact  m'  will  be  moving  faster  than  m. 
Let  the  velocity  of  m  after  impact  be  v  and  let 
that  of  m'  be  v'.  Then  v'>v.  During  the 
short  time  of  contact  each  body  exerts  a  force 
of  fpherefb^e°and  <>n  the  other  and,  by  the  Third  Law  of  Motion, 
after  impact.  these  forces  are  at  any  moment  equal  and 

opposite.  These  forces  also  act  for  the  same 
length  of  time  and  must,  therefore,  produce  equal  and  opposite 
changes  of  momentum.  Hence  the  total  momentum  after  im- 
pact equals  the  total  momentum  before  impact,  or 

mv  -f  m  V  =  mu  +  m'u'  (1) 

If  the  problem  be  to  find  the  velocities  after  impact,  this  equa- 
tion will  not  suffice,  since  it  contains  two  unknown  quantities  v 
and  i/.  A  second  relation  between  v  and  v'  was  discovered  ex- 
perimentally by  Newton.  He  found  that  for  given  materials,  the 
ratio  of  the  speed  of  separation,  (v'  —  v),  to  the  speed  of  approach, 
(u— w'),  is  a  constant,  which  is  (at  least  very  nearly)  independ- 
ent of  the  masses  and  velocities  of  the  bodies  and  depends  only 
on  their  materials  and  the  direction  of  the  grain,  if  they  are  not 
isotropic.  This  constant  ratio  is  called  the  coefficient  of  restitu- 
tion. Denoting  it  by  e,  we  have 

v'-v 

— t=e 
u-u' 

v-.v>=-e(u-u>)     ^  (2) 

From  (1)  and  (2)  v  and  v'  can  be  calculated.  For  simplicity 
in  establishing  these  equations,  we  chose  the  case  in  which  all  the 
velocities  are  in  the  positive  direction,  but  they  are  algebraic 


PROPERTIES  OF  SOLIDS 


117 


equations  applicable  to  all  cases.     In  applying  them  care  must 
be  taken  to  give  proper  signs  to  the  velocities. 

Some  simple  deductions  may  readily  be  drawn.  When  e  is  zero,  as 
it  very  nearly  is  for  such  soft  substances  as  putty  and  lead,  we  see  from 
(2)  that  v  and  i/  are  equal,  or  the  bodies  do  not  separate  after  impact. 

If  the  masses  of  the  spheres  be  equal  and  e  —  1,  the  spheres  will  on  impact 
exchange  velocities.  For  in  this  case  the  two  equations  become 


v  —  1/=  —  u+u' 

Hence  v  —  ur  and  v'  =  u,  which  proves  the  statement. 

If  one  of  the  bodies,  say  m',  is  of  very  great  mass  compared  with  the 
other  and  is  initially  at  rest,  its  velocity  after  impact  will  be  very  small. 
Putting  both  u'  and  v'  equal  to  zero  in  (2)  we  get 

LJ  .__; 

U 

This  is  the  case  when  a  small  ball  is  dropped  on  a  very  large  block.  Let 
the  height  of  fall  be  H  and  the  height  of  rebound  h.  Then  u=*\/t2gH 
downward  and  v  =«  \/2gh  upward.  Hence 


This  affords  a  simple  experimental  method  for  finding  e. 

176.  Oblique  Impact  of  Smooth  Spheres. — The  impact  of  two  spheres  is 
described  as  oblique,  when  the  spheres  are  not  moving  before  impact  in  the 
direction  of  the  line  through  their  centers.  The  lines  of  motion  of  the 
centers  before  impact  may  be  in  one  plane,  as  when  two  f 

equal  balls  rolling  on  a  plane  surface  impinge,  or  these 
lines  may  be  in  different  planes.  In  either  case  we  may 
resolve  the  velocity  of  each  ball  before  impact  into  two 
components,  one  in  the  direction  of  the  line  through 
the  centers  at  the  moment  of  impact,  the  other  in  a 
direction  perpendicular  to  that  line.  Only  the  first 
component  will  be  affected  by  the  impact  since  (the 
spheres  being  supposed  frictionless)  the  only  force  will 
be  a  pressure  in  the  line  of  the  centers.  The  change 
in  this  component  may  be  calculated  as  in  the  case  of  direct  impact;  then, 
by  compounding  this  component  for  each  sphere  after  impact  with  the 
unchanged  component,  we  can  find  the  motion  of  each  sphere  after  impact. 

When  a  smooth  ball  impinges  obliquely  with  a  velocity  u  on  a  fixed 
surface  in  a  direction  making  an  angle  a  with  the  normal,  its  component 
velocity  parallel  to  the  surface  is  u  sin  a  and  perpendicular  to  the  surface 
u  cos  a.  If  it  rebounds  with  avelocity  v  in  a  direction  making  an  angle  b 
with  the  normal,  the  components  become  v  sin  6  and  t;ct>s  6.  The  component 


Fia.  83. — Impact 
on  a  fixed  surface. 


118        MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

parallel  to  the  surface  is  not  changed,  while  that  perpendicular  to  the 
surface  is  changed  in  the  ratio  e  :  1.  Hence 

v  sin  b  =  u  sin  a,     v  cos  b=eu  cos  a. 
Hence,  by  dividing  corresponding  sides, 

,     tan  a 
tanfc-  — 
e 

Thus  the  direction  of  rebound  is  more  nearly  parallel  to  the  surface  than 
that  of  impact.  This  is  the  basis  of  a  method  that  has  been  employed  for 
finding  «. 

176.  Loss  of  Energy  on  Impact. — The  kinetic  energy  of  two 
smooth  spheres  before  impact  and  that  after  impact  can  be  calcu- 
lated from  their  masses  and  velocities.     The  total  kinetic  energy 
of  two  bodies  is  less  after  impact  than  before  (except  when  e  is 
unity)  and  other  forms  of  energy,  such  as  heat  and  sound,  are 
produced. 

177.  Vibration  of  Elastic  Bodies. — When  a  body  is  strained 
within  the  limit  of  elasticity,  the  internal  stresses  tend  to  restore 
the  body  to  its  original  condition.     When  released  from  the  ex- 
ternal deforming  force  the  body  vibrates,  and,  since  the  restoring 
forces  are  at  each  stage  proportional  to  the  distortion,  the  vibra- 
tions are  simple  harmonic  vibrations  of  a  constant  period.     This, 
for  instance,  is  the  case  when  a  rod  firmly  clamped  at  one  end 
is  bent  and  released.     When  the  vibrations  are  sufficiently  rapid, 
as  is  the  case  of  the  prongs  of  a  steel  tuning  fork,  sound  is  pro- 
duced, and  the  ear  can  test  constancy  of  the  period  of  vibra- 
tion by  the  steadiness  of  the  pitch;  the  vibrations  gradually  die 
down,  that  is,  the  extent  of  the  maximum  strain  in  each  vibration 
decreases,  yet  the  period  remains  unchanged,  showing  that  within 
the  limits  of  vibration  the  stress  is,  so  far  as  the  delicate  sense 
of  hearing  can  detect,  accurately  proportional  to  the  strain.     A 
tuning  fork  can  be  made  of  any  metal,  of  wood  or  other  solid 
substance;  and,  while  the  sound  may  in  many  cases  be  weak 
and  short-lived,  the  steadiness  of  pitch  while  it  lasts  is  an  ex- 
cellent proof  of  Hooke's  law. 

178.  Strain  Beyond  the  Elastic  Limit. — As  an  illustration  of 
what  happens  when  a  substance  is  strained  beyond  the  elastic 
limit,  that  is,  beyond  the  range  in  which  Hooke's  law  holds,  we 
shall  consider  the   stretching  of  a  wire.     When    a  force   that 


PROPERTIES  OF  SOLIDS  119 

stretches  it  beyond  the  limit  is  applied  to  it  and  this  force  is 
steadily  increased,  it  elongates  in  greater  proportion  for  each 
successive  equal  increase  of  the  force.  As  the  force  is  increased, 
at  a  certain  strain,  called  the  yield  point,  a  very  rapid  increase  of 
strain  sets  in  at  some  point  of  the  wire,  and  the  strain  at  that 
point  continues  to  increase,  even  if  the  force  is  not  increased, 
until  at  last  the  specimen  "necks  in"  and  breaks.  Beyond  the 
yield  point  the  substance  flows  much  like  a  very  viscous  liquid. 
If  during  this  process  the  force  be  diminished  somewhat,  the 
strain  will  still  continue  to  increase,  but  at  a  diminished  rate; 
and,  when  the  force  is  diminished  sufficiently,  the  strain  ceases 
to  increase  before  breaking  occurs.  If  at  this  stage  the  applied 
force  be  removed  entirely,  the  wire  will  contract  somewhat,  but 
a  large  permanent  set  will  remain.  The  wire  will  then  act  like  a 
different  wire  with  a  new  elastic  limit. 

179.  Elastic  After-effects. — From  strain  within  the  elastic  limit 
the  strained  material  completely  recovers  in  time  and  there  is  no 
permanent  set;  but  frequently  the  immediate  recovery  on  removal 
of  the  force  is  not  complete,  and  there  remains  a  small  temporary 
set  from  which  the  material  only  slowly  recovers.     This  slow 
recovery  from  temporary  set  is  called  an  elastic  after-effect.     It 
is  shown  by  rubber  and  glass  and  other  substances  which  consist 
of  mixtures  of  diverse  molecules;  but  crystals  and  quartz  threads 
do  not  show  it. 

It  is  readily  demonstrated  by  clamping  both  ends  of  a  rubber  cord  (used 
for  tires  of  small  wheels)  and  attaching  a  small  mirror  to  the  middle  to 
reflect  a  beam  of  light  on  a  scale.  Such  an  arrangement  will  show  a 
double  after-effect  due  to  successive  twists  in  opposite  directions. 

180.  Fatigue  of  Elasticity. — The  vibrations  of  a  torsional  pen- 
dulum are  maintained  by  the  elasticity  of  the  wire;  they  slowly 
die  away,  owing  to  air  resistance  and  internal  friction  in  the  wire. 
If  the  pendulum  be  by  some  means  kept  vibrating  a  long  time 
and  then  released,  the  vibrations  will  die  away  more  rapidly 
than  before,  as  if  the  elasticity  had  become  somewhat  exhausted 
by  prolonged  exercise.     This  fatigue  will  persist  for  a  long  time 
but  the  wire  will  promptly  recover  after  being  heated  to  about 
100°C. 

181.  Miscellaneous  Properties  of  Solids. — There  are  many    mechanical 
properties  of  solids,  frequently  mentioned,  which  are  not  yet  defined  with 


120        MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

sufficient  clearness  to  make  it  possible  to  measure  them,  but  which  call 
for  some  mention. 

A  malleable  body  is  one  which  can  be  hammered  into  thin  sheets.  The 
most  malleable  substance  is  gold,  which  can  be  reduced  to  sheets  of  gold 
foil  1/250,000  inch  in  thickness. 

A  ductile  substance  is  one  which  can  be  drawn  out  into  fine  wires.  Silver 
and  copper  are  very  ductile;  wires  less  than  1/1000  inch  in  diameter  are 
readily  made  from  these  metals.  By  heating  a  substance  until  it  is  semi- 
liquid  and  then  drawing  it  out,  fine  threads  of  substances  not  ordinarily 
ductile  can  be  made.  Fine  tubes  and  threads  of  glass  are  obtained  in  this 
way  and  fine  threads  of  quartz,  called  quartz  fibers,  are  thus  made  for 
use  in  suspensions  of  galvanometers  and  other  instruments;  they  enabled 
Boys  to  greatly  reduce  Cavendish's  apparatus  (§146). 

A  plastic  substance  is  one  which  can  be  moulded  by  pressure.  Many 
substances  not  ordinarily  regarded  as  plastic  are  so,  when  subjected  to 
great  pressure  slowly  applied.  A  stick  of  sealing  wax  is  ordinarily  brittle, 
but,  suspended  horizontally  on  end  supports,  it  will  slowly  yield  to  its 
weight  and  bend.  All  metals  under  enormous  shearing  stresses  become 
plastic.  The  impact  of  a  cannon  ball  on  armor-plate  will  sometimes 
produce  a  splash  like  a  stone  dropped  in  water. 

A  friable  substance  is  one  easily  reduced  to  powder  by  a  blow.  Glass, 
diamond  and  crystals  are  friable. 

Hardness  is  a  term  used  in  different  senses.  It  sometimes  means  the 
opposite  of  plasticity,  that  is,  resistance  to  change  of  shape,  as  when  we 
speak  of  iron  as  hard  and  rubber  as  soft.  Another  use  of  it  is  to  denote 
power  of  scratching,  as  in  the  mineralogists'  scale  of  hardness,  which  con- 
sists of  a  series  of  substances,  with  diamond  at  one  end  and  talc  at  the 
other,  arranged  so  that  each,  beginning  with  diamond,  will  scratch  the 
following  but  not  the  preceding.  Any  other  substance  that  will  scratch 
one  in  the  list  but  not  the  next  higher  is  said  to  have  a  hardness  between 
the  two. 

PROPERTIES  OF  FLUIDS 

182.  A  fluid  is  distinguished  from  a  solid  by  the  absence  of  per- 
manent resistance  to  forces  tending  to  produce  a  change  of  shape; 
that  is  to  say,  the  shear  modulus  of  a  fluid  is  zero.  In  this  respect 
all  fluids  agree;  they  also  agree  in  having  weight  and  inertia. 
Because  of  agreement  in  these  respects  there  are  certain  prop- 
erties common  to  all  fluids. 

In  certain  other  respects  liquids  and  gases  differ  considerably. 
These  differences  are  due  to  the  fact  that,  while  the  particles  of 
liquids  are  comparatively  close  together  and  attract  one  another 
with  very  considerable  forces,  the  particles  of  gases  are  so  far 
apart  that  the  forces  between  them  are  usually  negligible  (except 


PROPERTIES  OF  FLUIDS  121 

at  impact).     Properties  in  which  liquids  and  gases  differ  will 
therefore,  be  treated  in  separate  chapters. 

183.  Direction  of  Force  on  the  Surface  of  a  Fluid. — When  a 
fluid  is  at  rest,  the  force  acting  on  its  surface  must  be  perpen- 
dicular to  the  surface.     This  results  from  the  fact  that  the  shear 
modulus  is  zero;  for,  if  the  force  were  not  at  right  angles  to  the 
surface,  it  might  be  resolved  into  a  component  perpendicular  to 
the  surface  and  a  component  along  the  surface.     The  latter  would 
produce  a  sliding  motion,  or  a  shear  of  the  liquid  near  the  surface, 
so  that  the  liquid  could  not  be  at  rest. 

At  the  surface  of  contact  of  a  fluid  and  a  solid,  for  example 
at  any  part  of  the  surface  of  a  vessel  in  which  the  fluid  is  con- 
tained, the  force  exerted  by  the  fluid  on  the  solid  is  at  right  angles 
to  the  common  surface.  If  it  were  not,  the  reaction  of  the  solid 
on  the  fluid,  being  equal  and  opposite  to  the  force  of  the  fluid 
on  the  solid,  would  not  be  at  right  angles  to  the  surface  of  the 
fluid. 

At  the  free  surface  of  a  liquid,  that  is,  where  the  liquid  is  in 
contact  with  a  gas,  the  pressure  between  the  two  must  be  at  right 
angles  to  the  surface.  The  force  of  gravity  must  also  be  at 
right  angles  to  the  free  surface  of  a  liquid  at  rest  or  sliding 
motion  would  result.  Hence  the  free  surface  of  a  liquid  at  rest 
is  horizontal  unless  it  is  acted  on  by  some  other  force  than  gravity 
and  gas  pressure,  such  as  surface  tension  (which  we  shall  con- 
sider later)  or  magnetic  force  acting  on  a  magnetic  liquid. 

184.  Pressure  in  a  Fluid. — In  a  fluid  there  are  forces,  actions 
and  reactions  between  contiguous  parts  of  the  fluid.     These  forces 
are  due  to  several  causes.     The  weight  of  the  upper  layers  of  the 
fluid  has  to  be  sustained  by  the  lower  layers  and  a  pressure  thus 
results.     Force  on  the  surface  of  the  fluid,  if  it  be  completely 
enclosed,  produces  a  pressure  in  the  fluid;  this  is  true  not  only  of  a 
fluid  in  a  vessel  which  it  completely  fills,  but  also  of  a  liquid  the 
free  surface  of  which  receives  the  pressure  of  the  atmosphere  or  of 
any  gas  above  the  liquid.     (Another  cause  of  pressure  in  a  fluid 
is  referred  to  in  §206.) 

The  total  force  exerted  by  a  fluid  on  any  surface  is  called  the 
thrust  on  that  surface.  The  thrust  per  unit  of  area  at  a  point  on 
the  surface  is  called  the  pressure  intensity  or  simply  the  pressure 
at  that  point.  The  pressure  over  a  surface  may  be  either  uniform, 


122        MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

that  is,  the  same  at  every  point,  or  variable.  When  uniform  the 
pressure  at  any  point  equals  the  force  on  any  unit  of  area;  when 
variable  it  equals  the  average  pressure,  that  is,  the  force  on  an 
area  divided  by  the  area,  when  the  area  is  reduced  without  limit. 
Whatever  the  causes  of  pressure  in  a  fluid,  the  pressure  at  a 
point  is  the  same  in  all  directions,  that  is  to  say,  if  we  suppose 
an  imaginary  surface  to  separate  the  fluid  at  a  point  into  two 
parts,  the  pressure  of  each  part  on  the  other  is,  as  we  have  already 
seen,  perpendicular  to  this  surface,  and  it  is  also  the  same  no 
matter  how  the  imaginary  surface  is  supposed  to  be  inclined.  This 
is  nearly  obvious  from  the  mobility  of  the  fluid,  but  the  rigorous 
proof  of  the  statement  is  not  difficult. 

Let  0  be  the  point  considered  and  let  RO  and  R'O  be  any  two  directions 
through  0.  Around  0  suppose  a  small  prism  described,  and  let  two  of 
its  faces,  of  which  AB  and  AC  are  the  traces,  be  perpen- 
dicular to  RO  and  R'O  respectively;  while  the  third  face,  of 
which  BO  is  the  trace,  is  equally  inclined  to  AB  and  AC, 
and  let  the  ends  of  the  prism  be  planes  parallel  to  ABC. 
The  fluid  within  the  prism  is  at  rest  and  therefore  (neglect- 
ing its  weight  for  a  reason  stated  later)  the  thrusts  on  all 
its  faces  form  a  system  of  forces  in  equilibrium.  Hence  the 
sum  of  the  components  of  the  forces  in  the  direction  BC 
equals  zero.  Two  only  of  the  thrusts  have  components  in  the  direction  BC, 
namely,  those  on  A  B  and  AC.  Let  these  be  R  and  Rr  respectively.  They 
are  equally  inclined  to  BC,  and  if  each  makes  with  BC  the  acute  angle  0, 

R  cos  Q-Rr  cos  0  =  0 

Now  the  areas  of  the  faces  AB  and  AC  are  equal:    suppose  each    is  a. 
Cancelling  cos  0  and  dividing  by  a,  we  get 


If  we  now  suppose  the  prism  to  become  indefinitely  small,  R/a  becomes 
the  pressure  at  0  in  the  direction  RO  and  R'/a  becomes  the  pressure  at 
0  in  the  direction  R'O.  Since  RO  and  R'O  stand  for  any  directions 
through  0,  the  pressure  is  the  same  in  all  directions. 

As  stated  above  the  weight  of  the  prism  was  neglected.  As  the  prism  is 
diminished  without  limit,  the  weight  of  the  liquid  in  it,  which  is  propor- 
tional to  the  cube  of  its  dimensions,  decreases  more  rapidly  than  the  thrusts, 
which  are  proportional  to  the  squares  of  the  dimensions;  each  time  the 
prism  is  reduced  to  one-half  in  linear  dimensions  the  area  of  each  face  is 
reduced  to  one-fourth  and  the  weight  of  the  contained  liquid  is  reduced  to 
one-eighth.  Hence  when  the  prism  is  taken  small  enough  the  weight 
becomes  negligible  compared  with  the  thrusts 


PROPERTIES  OF  FLUIDS 


123 


US 


Fia.  85. 


185.  Pressure  at  Different  Points  in  a  Fluid.— (1)  Let  P  and  Q 

be  two  points  in  a  fluid  at  rest,  the  positions  of  the  points  being 

such  that  the  straight  line  PQ  is  horizontal  and  wholly  in  the 

fluid.     Consider  the  forces  acting  on  a  cylinder 

of  the  fluid  described  about  PQ  as  axis.     The 

thrusts  on  the  curved  surface  of  the  cylinder 

have  no  components  in  the  direction  of  the 

axis.     Hence,  for  equilibrium,  the  thrusts  on 

the  ends  must  be  equal  and  opposite;   and, 

since  the  ends  are  of  the  same  area,  the  average  pressures  on 

the  ends  must  be  equal.     If,  now,  the  radius  of  the  cylinder  be 

supposed  indefinitely  decreased,  the  average  pressures  on  the 
ends  become  the  pressures  at  P  and  Q,  which  must, 

therefore,  be  equal.     Hence  the  pressure  in  any  direc- 
tion at  P  equals  the  pressure  in  any  direction  at  Q. 

(2)  Let  P  and  Q  be  two  points  in  a  vertical  line 
wholly  in  a  fluid  of  density  p.  Consider  the  forces 
acting  vertically  on  a  cylinder  described  with  PQ  as 
axis  and  of  unit,  1  sq.  cm.,  cross-section.  If  the  depth 
of  Q  below  P  be  h  cms.  the  volume  of  the  cylinder  will 
be  h  c.c.,  its  mass  will  be  hp  gms.  and  its  weight  hpg 
dynes.  If  pl  be  the  pressure  in  dynes  per  sq.  cm.  at 

P  and  p2  that  at  Q,  the  thrust  downward  at  P  will  be  pl  and 

that  upward  at  Q  will  be  p2.     Hence 


Fio.  86. 


(3)  Let  P  and  Q  be  any  two  points  in  the  fluid. 
No  matter  what  the  shape  of  the  containing 
vessel,  P  and  Q  can  be  connected  by  a  broken 
line  made  up  of  vertical  and  horizontal  steps. 
•Along  the  zigzag  path  from  P  to  Q  there  will  be 
a  difference  of  pressure  h'pg  for  each  vertical 
step  of  length  h',  while  for  each  horizontal  step 
there  will  be  no  change  of  pressure.  Hence  the 
difference  of  pressure  between  P  and  Q  will  be 
gp  X  (the  algebraic  sum  of  the  vertical  steps)  or,  if  the  difference 
of  level  of  P  and  Q  be  h,  the  difference  of  pressure  will  be  hpg. 

186.  Pressure  in  a  Gas. — Since  the  density  of  a  gas  is  compara- 
tively small,  the  difference  at  two  points  is  usually  so  slight 


Fia.  87. 


124        MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

as  to  be  negligible;  but  this  is  not  the  case  if  h  be  very  great.  Thus 
in  a  vessel  containing  gas  the  pressure  may  be  regarded  as  every- 
where the  same;  but  the  pressure  of  the  air  varies  greatly  as  we 
ascend  to  great  heights  in  the  atmosphere  or  descend  to  great 
depths  in  a  mine. 

187.  Units  Employed  in  Calculating  Fluid  Pressure.  —  In  estab- 
lishing the  formula  for  difference  of  pressure  at  different  depths 
in  a  fluid,  namely, 


it  has  been  supposed  that  absolute  units  are  employed.  If  h 
be  in  cms.,  p  ingms.  per  c.c.  and  g  in  cm.  per  sec.2  (about  980), 
pl  and  p2  will  be  in  dynes  per  sq.  cm.  A  dyne  per  sq.  cm.  is 
sometimes  called  a  bar. 

When  the  values  of  p±  and  p2  would  be  inconveniently  large 
in  absolute  units,  other  units  may  be  employed.  If  g  be  omitted, 
pl  and  p2  will  be  in  gms.  wt.  per  sq.  cm.  and 

P»-Pi=V 

This  formula  may  also  be  used  to  calculate  the  pressure  in  metric 
tons  (1,000,000  gms.)  per  sq.  m.  (10,000  sq.  cm.)  if  h  be  in 
meters  (100  cm.). 

When  British  units  are  employed  the  weight  of  a  cylinder  of  1 
sq.  ft.  cross-section  and  h  feet  in  length  and  of  density  p  (Ibs. 
per  cu.  ft.)  is  hp  Ibs.  Hence  if  pl  and  p2  are  in  Ibs.  wt.  per  sq.  ft., 

P2-Pi=V 

188.  Surface  of  Contact  of  Two  Fluids.  —  The  surface  of  con- 
tact of  two  fluids  of  different  densities  which  are  at  rest  and  do 
not  mix  is  horizontal.     This  may  be  deduced 
from  the  principle  that,  for  stable  equilibrium, 
the  potential  energy  of  a  system  must  be  a 


FIQ  88_  Thesur-    mmimum  (§107).     If   any  part  of  the  denser 
ace  of  contact  of  two    fluid  were  at  a  higher  level  than  an  equal  part 
asTaf  is*         °f  *ke  IGSS  dense,  ^ne  potential  energy  could  be 


decreased  by  interchanging  the  two.  Hence,  for 
the  potential  energy  to  be  a  minimum,  every  part  of  the  denser 
fluid  must  be  lower  than  any  part  of  the  less  dense,  that  is,  the 
surface  of  contact  must  be  horizontal  with  the  denser  liquid 
below.  Another  proof  is  to  suppose  that  the  surface  could  be 
inclined,  as  LM.  Let  P  and  Q  be  two  points  in  the  surface. 


PROPERTIES  OF  FLUIDS 


125 


Complete  a  rectangle  AQBP  with  vertical  and  horizontal  sides. 
The  pressure  at  A  would  equal  that  at  P  and  the  pressure  at  Q 
would  equal  that  at  B.  The  increase  of  pressure  from  A  to  Q 
would  equal  the  increase  from  P  to  B  and  this  could  not  be  when 
the  liquids  are  of  different  density. 

A  particular  case  of  the  above  is  the  surface  of  contact  of  a 
liquid  with  the  atmosphere  or  any  gas;  it  must  be  horizontal. 

In  both  proofs  it  has  been  assumed  that  gravity  is"  the  only  force 
acting  on  the  particles  of  the  fluids;  if  any  other  force  exist,  the 
surface  may  not  be  horizontal.  In  any  case  it  is  at  right  angles 
to  the  resultant  force. 

189.  Pascal's  Principle. — When  a  fluid  is  at  rest,  the  difference 
of  pressure  between  two  points  depends  only  on  the  difference  of 
level  and  the  density  (§185) .  Hence,  if  the  pressure  at  any  point 
be  increased,  there  will  be  an  equal  increase  of  pressure  at  every 
point  (provided  the  density  does  not  change  appreciably)  or  pres- 
sure is  equally  transmitted  in  all  directions.  This  is  Pascal's 
principle  of  the  transmissibility  of  pressure. 


Fio.  89. — Hydraulic  press. 


Pascal's  principle  is  not  rigorously  true  for  a  compressible 
fluid,  for  pressure  will  produce  a  change  of  density  of  a  com- 
pressible fluid.  But  the  compressibility  of  liquids  is  so  small 
that  the  principle  is  practically  true  for  all  liquids.  Gases  are 
much  more  compressible,  but  their  densities  under  ordinary 


126        MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

circumstances  are  so  small  that  the  pressure  in  a  moderate  volume 
is  everywhere  practically  the  same  and  the  principle  is  practically 
true  for  gases  also. 

190.  The  Hydraulic  Press. — In  the  hydraulic  press  Pascal's 
principle  is  applied  to  obtain  a  great  force  by  the  exertion  of  a 
relatively  small  one.     It  consists  of  a  large  cylinder  and  piston 
(or  plunger)  and  a  small  cylinder  and  piston,  the  two  cylinders 
being  connected  by  a  tube  and  filled  with  some  liquid.      Let 
the  area  of  the  large  piston  be  £  and  that  of  the  small  one  s.     If 
the  pressure  in  the  liquid  is  p,  the  thrusts  on  the  pistons  are  pS 
and  ps  respectively,  and  these  are  in  the  ratio  of  S  :  s.     Hence 
a  small  external  force  applied  to  the  small  piston  will  enable  the 
large  piston  to  exert  a  relatively  great  external  force.  The  arrange- 
ment is  indicated  in  figure  88;  a  valve  in  the  connecting  tube 
permits  flow  from  the  smaller  cylinder  toward  the  larger,  but 
not  in  the  opposite  direction. 

191.  Archimedes'  Principle. — When  a  body  is  partly  or  wholly 
immersed  in  a  fluid  at  rest,  every  part  of  the  surface  in  contact 
with  the  fluid  is  pressed  on  by  the  latter,  pressure  being  greater 
on  the  parts  more  deeply  immersed.     The  resultant  of  all  these 
forces  of  pressure  is  an  upward  force  called  the  buoyancy  of  the 
body  immersed.     The  direct  calculation  of  this  resultant  force  is 
difficult  except  when  the  body  has  some  simple  form,  such  as  a 
cylinder  with  its  axis  vertical;  but  a  simple  line  of  reasoning  will 
show  the  magnitude  and  direction  of  the  force. 

The  pressure  on  each  part  of  the  surface  of  the  body  is  evi- 
dently independent  of  the  material  of  which  the  body  consists. 
So  let  us  suppose  the  body,  or  as  much  of  it  as  is  immersed,  to 
be  replaced  by  fluid  like  the  surrounding  mass.  This  fluid  will 
experience  the  pressures  that  acted  on  the  immersed  body  and  this 
fluid  will  be  at  rest;  hence  the  resultant  upward  force  on  it  will 
equal  its  weight  and  will  act  vertically  upward  through  its  center 
of  gravity.  It  follows  that  a  body  wholly  or  partly  immersed  in 
a  fluid  is  buoyed  up  with  a  force  which  is  equal  to  the  weight  of 
the  volume  of  the  fluid  which  the  body  displaces  and  which  acts 
vertically  upward  through  the  center  of  gavity  of  the  fluid  be- 
fore its  displacement.  This  point  through  which  the  force  of 
buoyancy  acts  is  called  the  center  of  buoyancy. 

Since  the  weight,  in  dynes,  of  the  fluid  displaced  equals  the 


PROPERTIES  OF  FLUIDS  127 

product  of  the  volume  (which  equals  the  volume  immersed)  its 
density  and  g, 

Buoyancy  —  Volume  immersed  X  density  of  fluid  X  g 

Buoyancy  is  to  be  treated  as  any  other  force  that  acts  on  a  body 
and  either  causes  motion  or  helps  to  produce  equilibrium.  If  a 
body  of  mass  M,  wholly  or  partly  immersed  in  a  fluid,  be  sustained 
partly  by  buoyancy  and  partly  by  another  upward  force  F,  then, 
in  absolute  units 


where  V  is  the  volume  immersed  and  p  is  the  density  of  the  fluid. 
When  gravitational  units  are  employed  g  must  be  omitted. 

192.  Fluids  in  Motion.  —  While  the  calculation  of  the  motion  of  a 
rigid  solid  body  is  comparatively  simple,  owing  to  the  fact  that 
we  may  treat  a  solid  as  a  whole  without  regard  to  the  actions 
between  its  parts,  the  discussion  of  the  motion  of  a  fluid  is  ren- 
dered difficult  by  the  readiness  with  which  any  part  of  the  fluid 
changes  its  shape,  and  we  cannot,  therefore,  without  the  use  of 
advanced  mathematics,  treat  of  any  except  a  very  few  and  simple 
cases  of  the  motion  of  fluids. 

When  a  fluid  moves  either  in  an  open  stream  or  in  a  closed 
pipe,  the  continual  change  of  shape  of  each  part  is  opposed  by 
internal  friction  between  these  parts,  and  to  maintain  the  motion 
some  external  force  must  be  applied  to  the  fluid.  The  most  com- 
mon causes  of  motions  of  fluids  are  gravity,  as  in  the  case  of  a 
river,  pressure  applied  to  some  part  of  the  boundary  of  the  fluid, 
as  in  the  case  of  water  pumped  through  a  system  of  piping,  and 
the  motion  of  solids  in  contact  with  the  fluid,  as  in  the  case  of 
a  fan. 

193.  Flow  in  Pipes.  —  When  the  pressure  on  a  fluid  in  a  hori- 
zontal tube  is  greater  at  one  end  than  at  the  other,  a  flow  ensues. 
When  the  pressure  is  first  applied,  the  motion  begins  with  an 
acceleration,  but  after  a  time,  if  the  pressure  at  the  ends  are  kept 
constant  and  the  supply  of  fluid  is  maintained,  a  steady  state  of 
motion  ensues,  so  that  at  each  part  of  the  tube  the  motion  is  con- 
stant.    The  simplest  case  is  when  the  tube  is  of  constant  cross- 
section  and  the  fluid  is  pratically  incompressible,  that  is,  a  liquid. 
In  this  case  the  volume  of  fluid  passing  all  cross-sections  of  the 


128        MECHANICS  AND  THE  PROPERTIES  OF  MATTER 


Fio.  90. 


pipe  is  the  same  throughout,  and  the  rate  of  flow  is,  therefore, 
the  same  at  each  cross-section.  The  motion  is  from  places  of 
higher  pressure  to  places  of  lower  pressure.  If,  however,  the 
fluid  be  compressible,  while  the  motion  at  a  point  remains  steady 
and  the  mass  that  passes  every  cross-section  is  necessarily  the 
same  as  that  which  enters  the  pipe,  the  volume 
of  flow  is  variable;  for  where  the  pressure  is 
greater  the  fluid  is  compressed  into  a  smaller 
volume,  and  where  the  pressure  is  less  the  fluid 
is  not  so  much  compressed.  Thus  the  speed  of 
the  fluid  particles  is  on  the  whole  greater  in  the  parts  of  the 
pipe  where  the  pressure  is  less,  that  is,  the  further  along  the 
stream  we  go. 

When  a  liquid  flows  through  a  tube  of  variable  section  (Fig.  90), 
the  pressure  at  the  ends  being  constant,  the  mass  that  passes 
each  cross-section  is  the  same,  but  the  rate  of  motion  of  the  par- 
ticles increases  as  the  stream  comes  to  a  contraction  of  the  tube,  and 
decreases  again  as  the  stream  comes  to  an  expansion  of  the  tube. 
Now  an  increase  of  velocity  or  an  acceleration  necessarily  means 
a  smaller  pressure  ahead  than  behind,  and  a  de- 
crease of  velocity  necessarily  means  a  larger 
pressure  ahead  than  behind.  Thus  in  a  con- 
traction (or  "throat")  the  pressure  is  smaller 
than  immediately  before  or  behind,  the  amount 
of  difference  being  dependent  on  the  rate  of 
flow  through  the  tube  and  the  cross-section  at 
the  throat  and  at  either  side.  This  principle  is 
the  basis  of  the  Venturi  meter  for  gauging  the 
flow  of  water  in  pipes.  The  same  principle  is 
employed  in  the  aspirator  (Fig.  91),  a  form  of 
air-pump  attached  to  a  water  faucet.  The 
water  is  forced  through  a  contraction  in  a  tube 
and  produces  suction  in  a  side  tube  which  is 
connected  to  the  vessel  to  be  exhausted.  Similar  considera- 
tions apply  to  the  flow  of  gas  through  a  pipe  of  variable  cross- 
section,  but  this  case  is  complicated  by  the  changes  of  volume 
due  to  changes  of  pressure. 

194.  Illustrations   of    the   Above. — Fig.  92    represents    (in    section)    a 
glass  tube   that  passes  tightly  through   a  wide   cork  and  a  second  cork 


Fia,  Qi. 


PROPERTIES  OF  FLUIDS 


129 


Fia.  92. 


through  the  center  of  which  a  pin  is  stuck.  When  air  is  blown  through 
the  tube  the  lower  cork  is  not  repelled  but  is  attracted  (the  pin  prevents 
side  motion).  The  air  increases  its  speed  to  pass  through 
the  small  space  between  the  corks,  hence  its  pressure  di- 
minishes and  atmospheric  pressure  pushes  the  corks  to. 
gether.  Various  other  pieces  of  apparatus,  such  as  the 
atomizer,  the  ball  nozzle  and  the  injector,  act  on  the  same 
principle. 

The  curvature  of  the  path  of  a  rotating  base  ball  or 
tennis  ball  is  due  to  a  difference  of  pressure  on  the  oppo- 
site sides  of  the  ball.  Suppose  the  ball  had  no  translatory  motion  but 
had  a  motion  of  rotation,  while  a  current  of  air  blew  on  it  as  indicated 
in  figure  93.  The  rotating  ball  would  carry  air  around  with 
it.  At  A  the  two  air  motions  would  conspire  and  at  B  they 
would  be  in  opposition.  Hence  the  velocity  of  the  air-par- 
ticles would  be  greater  at  A  than  at  B  and  the  pressure  at 
B  would,  therefore,  be  greater  than  that  at  A,  the  result  being 
a  force  on  the  ball- in  the  direction  B  A.  The  same  result  fol- 
lows when  the  ball  is  moving  through  air  otherwise  at  rest, 
and  the  path  curves  toward  the  side  of  less  pressure.  The 
motion  of  the  ball  can  also  be  explained  by  considering  that 
the  impacts  between  the  ball  and  the  air  particles  are  neces- 
sarily more  violent  on  the  side  B  than  on  the  side  A. 


o 

Fio.  93.-=- 
" Curve"  of  a 
base  ball. 


Fio.  94. — Work  by  a 
piston. 


195.  Work  Done  by  a  Piston. — When  a  piston  of  area  a  moves 
a  small  distance  d  along  a  cylinder  against  a  pressure  P  (per  unit 
area),  it  exerts  a  force  Pa  through  a  dis- 
tance d,  and  therefore  does  work  Pad. 
Since  ad  is  the  volume,  say  J7,  of  the 
small  part  of  the  cylinder  through  which 
the  piston  has  moved,  the  work  done  is 
P.  4V. 

If  the  pressure  is  not  constant,  as,  for  example,  where  a  gas  in 
a  closed  vessel  is  being  compressed,  the  whole  work  will  be  the 
sum  of  products  P.  AV ,  where  P  must  be  given 
its   proper  value  for  each  successive  change  of 
volume  A  V.     We  may  also  use  a  graphical  method 
as  in  §22  and  §56.     In  the  present  case  (Fig.  95) 
each  abscissa  will  represent  the  volume  at  some 
moment  and  the  corresponding  ordinate  will  rep- 
resent the  pressure  at  that  moment.    The  area 
will  represent  the  whole  work. 
Conversely,  an  expanding  fluid  does  work  equal  to  the  sum  of 


V     AV      V 
FIG.  95. 


n 


130        MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

P.  AVy  where  P  is  the  pressure  and  4V  an  increment  of  volume. 
If  P  is  constant  and  the  whole  change  of  volume  is  V,  the  work 
done  is  PV. 

196.  Viscosity. — A  fluid  offers  no  permanent  resistance  to  forces 
tending  to  change  its  shape;  it  yields  steadily  to  the  smallest  de- 
forming force.  But  the  rate  of  yielding  is  different  for  different 
fluids,  that  is,  different  fluids  offer  different  transient  resistances 
to  deformation.  This  transient  resistance  is  called  internal  fric- 

tion  or  viscosity.     Thus  a  very  viscous  liquid 

..b         such   as   glycerine   or   tar  flows  much  more 
slowly   through   a  tube  or  down  an  incline 

— - — £_J! — £ than  water  does,  and  such  a  flow  consists  in  a 

Fl°'  967^dearin8  °f  a    continuous  change  of  shape  of  each  part  of  the 
liquid.     The  internal  forces  are  what  we  have 
called  stresses,  and,  since  the  strain  is  a  change  of  shape  only, 
the  stress  must  be  a  shearing  stress. 

Consider,  as  an  example,  a  stream  (Fig.  96)  flowing  down  a 
very  gentle  incline  under  the  force  of  gravity.  The  motion  is 
greater  near  the  surface  than  at  the  bottom.  A  small  cube 
A  BCD  with  sides  vertical  and  horizontal  will,  by  the  motion,  be 
changed  into  the  form  abed.  The  liquid  above  AB  exerts  a  force 
in  the  direction  A  B,  on  the  upper  face  of  the  cube,  and  the  liquid 
below  CD  exerts  a  resisting  force  on  the  face  CD  in  the  direc- 
tion CD.  These  two  forces  consti- 
tute a  shearing  stress.  A  similar  de-  ^/.^x^v^^^^^tft. 
scription  applies  to  a  small  cube  of  a  | ! 
liquid  flowing  in  any  manner  what- 


ever.     Very   extensive  experiments-  Flo-  97> 

have  shown  that  the  ratio  of  the  shear t 

ing  stress  to  the  rate  of  shear  is  a  cons  ant  for  any  one  fluid,  the 

value  of  the  constant  being  different  for  different  fluids.     This 

is  the  fundamental  and  very  simple  law  of  fluid  friction. 

The  constant  ratio  of  the  shearing  stress  in  a  fluid  to  its  rate 
of  shear  is  called  the  coefficient  of  viscosity  of  that  fluid. 

A  concrete  case  will  make  this  definition  clearer  and  will  lead 
to  another  way  of  stating  the  definition.  Suppose  (Fig.  97) 
that  the  space  between  two  large  parallel  plates  is  filled  with  the 
fluid  under  consideration  and  let  one  plate  be  moving  parallel  to 
the  other  with  a  velocity  v.  Experiment  (as  stated  later)  shows 


PROPERTIES  OF  FLUIDS  131 

that  the  fluid  in  contact  with  the  plates  does  not  slip  along  the 
faces  of  the  plates  but  adheres  to  them.  The  moving  plate  will 
in  a  very  short  time  t  move  a  distance  vt,  and,  if  the  distance 
between  the  plates  be  d,  the  shear  produced  in  the  time  t  will  be 
vt/d.  Hence  the  rate  of  shear  is  v/d.  If  the  area  of  each  plate 
be  A  and  the  force  applied  to  move  the  upper  plate  be  F,  the 
shearing  stress  will  be  F/A.  Hence,  denoting  the  coefficient  of 
viscosity  by  /j.  we  have 


v/d 

Av 


If,  now,  we  suppose  A,  v  and  d  to  be  each  unity,  F  will  be  equal 
to  fjL.  Hence  we  have  the  following  definition  of  /*:  The  coeffi- 
cient of  viscosity  of  a  fluid  is  the  tangential  force  on  unit  of 
area  of  either  of  two  horizontal  planes  at  the  unit  distance 
apart,  one  of  which  is  fixed  while  the  other  moves  with  the 
unit  velocity,  the  space  between  them  being  filled  with  the  viscous 
material. 

197.  Measurement  of  Coefficients  of  Viscosity.  —  The  most  com- 
mon method  of  finding  JJL  is  by  measuring  the  flow  of  the  fluid 
through  a  tube  of  very  small  bore  (or  so-called  capillary  tube). 
The  motion  of  the  fluid  in  such  a  case  (provided  the  velocity  does 
not  exceed  a  certain  magnitude)  is  analogous  to  the  slipping  of 
the  tubes  of  a  small  pocket  telescope  through  one  another.  If 
we  imagine  the  fluid  divided  into  a  very  large  number  of  thin 
cylindrical  shells,  the  motion  consists  in  the  slipping  of  shell 
through  shell;  hence  the  resistance  encountered  is  internal  fric- 
tion or  viscosity.  Let  p  be  the  difference  of  pressure  per  unit  area 
at  the  two  ends  of  the  tube  (supposed  horizontal)  ,  I  the  length  of 
the  tube,  and  r  its  radius.  It  has  been  shown  theoretically  and 
experimentally  that,  when  the  fluid  is  a  liquid,  the  volume  that 
flows  out  of  the  tube  in  unit  time  is 


This  formula  also  applies  to  a  gas  if  p  be  very  small,  but  if  p  be 
large  the  formula  must  be  modified  to  allow  for  the  compressi- 
bility of  the  gas.  In  the  theoretical  proof  of  the  above  formula 


132        MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

it  is  assumed  that  no  slipping  of  the  fluid  along  the  surface  of  the 
tube  takes  place,  and  the  agreement  of  theory  and  experiment 
confirms  this  assumption. 

The  following  are  some  values  of  /i  in  c.g.s.  units  at  20°C. 

Alcohol..: 0.0011     Water 0.010 

Ether 0.0026     Glycerin 8.0 

198.  The  Explanation  of  Viscosity. — Viscous  resistance  to  fluid  motion 
resembles  friction  between  solids  in  certain  respects,  and  in  other  respects 
the  two  are  very  different.  Both  are  forces  that  appear  only  as  resistances 
to  relative  motion;  they  are,  therefore,  non-conservative  forces  and  energy 
spent  in  doing  work  against  them  is  changed  into  heat.  But,  while  the 
friction  between  solids  is,  through  a  considerable  range  of  velocity,  inde- 
pendent of  the  velocity,  the  resistance  due  to  viscosity  is  exactly  propor- 
tional to  velocity  through  the  widest  range  in  which  experimental  tests 
have  been  made.  This  points  to  a  fundamental  difference  in  the  nature 
of  the  resistance  in  the  two  cases. 

There  are  many  strong  reasons  for  believing  that  the  particles  of  fluid  are 
in  rapid  motion  and  are  not,  like  the  particles  of  solids,  confined  to  more 
or  less  definite  positions.  If  now  we  imagine  two  layers  of  a  fluid  in  relative 
motion,  so  that  one  is  passing  another,  like  one  railway  train  passing  a 
second,  it  is  evident  that  particles  from  each  layer  must  be  continually 
crossing  the  boundary  into  the  other  layer.  The  particles  of  the  more 
rapidly  moving  layer  that  cross  the  boundary  carry  their  larger  momentum 
with  them  and  thus  produce  a  gradual  increase  of  the  velocity  of  the  second 
layer.  At  the  same  time  particles  of  the  latter  layer  penetrate  into  the 
former  and  by  taking  up  momentum  diminish  the  velocity  of  that  layer. 
The  result,  on  the  whole,  is  a  tendency  of  the  two  layers  to  come  to  the 
same  velocity,  and  this  is  exactly  what  we  mean  by  a  resistance  to  relative 
motion.  In  the  case  of  gases  this  explanation  may  be  regarded  as  fully 
established;  for  the  formulas  to  which  it  leads  by  mathematical  methods 
are  verified  by  experiment.  It  has  not  yet  been  found  possible  to  work 
out  the  mathematical  results  in  the  case  of  liquids,  but  there  is  no  reason 
to  doubt  that  the  explanation  is  equally  applicable  to  the  latter. 


Liquids 

199.  Compressibility  of  Liquids. — While  the  shear-modulus  of 
any  liquid  is  zero  the  bulk-modulus  is  usually  large,  that  is,  the 
pressure  on  a  liquid  must  be  greatly  increased  to  produce  much 
diminution  of  volume.  The  coefficient  of  compressibility  of  a 
liquid  (§169)  is  therefore  small.  Measurements  of  the  compres- 
sibilities of  liquid  are  made  by  subjecting  the  liquids  to  great 


PROPERTIES  OF  FLUIDS 


133 


pressures  in  a  vessel  called  a  piezometer  and  noting  the  resulting 
diminution  of  volume. 

The  following  table  gives  the  compressibilities  of  some  liquids. 
Each  number  is  the  proportion  by  which  the  volume  of  the 
liquid  is  decreased  when  the  pressure  on  it  is  increased  by  one 
atmosphere. 

Alcohol 0 . 0000828        Mercury 0 . 0000038 

Ether 0.0001156         Water. 0.0000489 

200.  Hydrometers. — A  hydrometer  is  an  instrument  for  finding 
the  density  of  liquids;  some  hydrometers  may  also  be  used  to 
find  the  density  of  solids.  The  action  of  most  hydrometers 
depends  on  Archimedes'  principle.  Some  hydrometers  sink  to 
different  depths  in  different  liquids  and  thus  indicate  the  densities 
of  the  liquids;  these  are  called  hydrometers  of  variable  immersion. 
Other  hydrometers  are  used  with  different  weights  added  to  the 


Fio.  98. — Common  hydrometer. 


Fia.  99. — Nicholson's  hydrometer. 


weight  of  the  instrument  so  that  they  are  always  immersed  to  the 
same  depth;  these  are  called  hydrometers  of  constant  immersion. 
The  common  hydrometer  is  one  of  variable  immersion.  It  is  a 
glass  tube  with  an  enlargement  in  the  middle  and  weighted  at  the 
lower  end  with  mercury  so  that  it  will  float  in  stable  equilibrium. 
Inside  the  tube  is  a  scale  which  indicates  the  density  of  the 
liquid  by  the  depth  to  which  the  tube  is  immersed  (Fig.  98). 


134        MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

The  best  known  hydrometer  of  constant  immersion  (Fig.  99)  is  Nicholson's 
Hydrometer.  It  consists  of  a  hollow  cylindrical  body  (of  metal  or  glass), 
to  one  end  of  which  a  somewhat  heavy  basket  B  is  attached,  while  at 
the  other  end  there  is  a  stem  S  which  carries  a  scale  pan  C  for  weights. 
On  the  stem  there  is  a  mark  indicating  the  depth  to  which  the  hydrometer 
is  to  -be  immersed.  Let  W  be  the  weight  of  the  hydrometer  and  let  w 
be  the  weight  that  must  be  placed  on  the  pan  to  make  the  instrument 
sink  to  the  mark  in  water  of  density  p,  and  uf  the  weight  on  C  required 
when  the  hydrometer  is  in  a  liquid  density  p'.  The  volume  of  liquid  dis- 
placed in  both  cases  is  the  same.  Hence,  by  Archimedes'  principle,  the 
weights  of  equal  volumes  of  the  second  liquid  and  of  water  are  W  +w'  and 
W+w,  and  their  ratio  is  the  ratio  of  the  densities. 

This  hydrometer  may  also  be  used  to  find  the  density  of  a  small  solid. 
When  so  used  the  instrument  is  in  reality  a  balance  for  weighing  the  solid 
in  air  and  then  in  some  liquid  of  known  density.  The  body  is  first  placed 
on  C.  The  weight  required  on  C  to  sink  the  hydrometer  to  the  mark  on 
the  stem  will  be  less  than  w'  by  the  weight  of  the  body.  This  gives  the 
weight  of  the  body  in  air.  The  body  is  then  placed  in  B  and  its  apparent 
weight  when  immersed  is  found  in  the  same  way.  The  ratio  of  the  weight 
of  the  body  to  its  apparent  loss  of  weight  when  immersed,  which  equals 
the  weight  of  an  equal  volume  of  liquid,  gives  the  specific  gravity  of  the 
body  relatively  to  the  liquid. 

201.  Stability  of  Flotation. — A  body  floating  at  rest  on  the  sur- 
face of  a  liquid  is  in  equilibrium  under  the  action  of  its  weight 
acting  vertically  downward  through  the  center  of  gravity,  G,  and 


•  Fio.  100.— Stable  equilibrium  of  a  vessel. 

the  resultant  upward  pressure  of  the  liquid  acting  through  the 
center  of  buoyancy,  B.  Hence  the  two  forces  are  equal  and  act 
in  opposite  directions  in  the  vertical  line  BG.  Suppose  the  body 
to  rotate  slightly  about  an  axis  perpendicular  to  the  plane  repre- 
sented in  the  figure.  The  form  of  the  volume  of  water  displaced 
is  now  different  (unless  the  body  be  spherical  or  cylindrical),  and 
the  center  of  buoyancy  is  at  some  point  B'  not  in  the  vertical  line 


PROPERTIES  OF  FLUIDS  135 

through  G.  Hence  the  forces  now  acting  on  the  body  constitute 
a  couple,  and,  if  the  couple  tends  to  right  the  body,  the  equilib- 
rium is  stable;  if  not,  it  is  unstable. 

The  simplest  case  to  consider  is  when  the  body  is  symmetrical,  or  very 
nearly  so,  on  opposite  sides  of  the  plane  through  B  and  G  perpendicular 
to  the  axis  of  rotation;  for  in  this  case  B'  is  in  this  plane.  Let  a  vertical 
through  B'  cut  BG  in  M  .  For  small  rotations  the  position  of  M  on  BG 
is  very  nearly  independent  of  the  magnitude  of  the  rotation.  M  is  called 
the  metacenter  of  the  body.  The  position  of  M  can  usually  be  calculated 
by  mathematical  methods.  If  M  is  above  G  it  is  evident  that  the  couple 
tends  to  right  the  body  and  the  equilibrium  is  stable;  if  M  is  below  G  the 
couple  tends  to  displace  the  body  further  and  the  equilibrium  is  unstable. 
Hence  the  danger  of  taking  the  whole  cargo  out  of  a  vessel  without  putting 
in  ballast  and  the  risk  of  upsetting  when  several  people  stand  up  at  once 
in  a  small  boat.  A  ship  has  one  metacenter  for  rolling  and  another  for 
pitching.  In  general  the  vessel  is  not  quite  of  the  symmetrical  form 
assumed  above  and  the  problem  of  stability  is  more  complicated.  (Article 
"Shipbuilding,"  Ency.  Britt.,  llth  ed.) 

202.  Energy  of  a  Moving  Stream.  —  When  liquid  flows  steadily  through 
a  pipe  of  varying  cross-section,  the  total  energy  in  the  space  between  any 
two  sections  A  and  B  remains  constant.  When  a  volume  V 
flows  in  through  A,  an  equal  volume  flows  out  through  B. 
Let  the  pressure  at  A  and  B  respectively  be  pl  and  p2, 
and  the  velocities  r,  and  va  respectively.  Let  p  be  the 
density  of  the  liquid.  When  the  volume  V  flows  in  through 
A  it  carries  kinetic  energy  JtVpVj*  into  the  space  between 
A  and  B  and  in  the  same  time  the  volume  V  flows  out 
through  B  carrying  energy  iFpv2a.  There  is  thus  a  gain 
of  kinetic  energy  \Vp(v*-v*).  ^  101 

Now  the  liquid  above  A  acts  Hke  a  piston  in  forc- 
ing liquid  into  the  space  between  A  and  B,  and  i*  thus  does  work 
piV  which  goes  to  increase  the  energy  between  A  and  B\  and  in  the 
same  time  the  liquid  between  A  and  B  does  work  p2F  in  forcing  liquid 
out  through  B.  From  this  cause  there  is  an  increase  of  energy  piV  —  p3V 
between  A  and  B.  If  between  A  and  B  there  is  a  fall  of  level  from  hl  to 
ha,  the  liquid  which  flows  in  at  A  will  have  a  greater  amount  of  potential 
energy  than  that  which  flows  out  at  B,  and  there  will,  therefore,  be  an 
increase  of  potential  energy  of  Vpg  (hl  —  h2)  between  A  and  B.  But  the 
total  energy  between  A  and  B  remains  constant.  Hence 


or 

Pi  +  gphi  +  ipVi8  —  p2  +  gph9  +  ipV  —  a  constant 

This  is  Bernoulli's  theorem.     It  is  of  great  importance  in  hydraulics. 


136        MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

203.  Outflow  from  an  Orifice— Torricelli's  Theorem. — When  an 
orifice  is  opened  in  a  side  of  a  vessel  containing  liquid  at  greater 
than  atmospheric  pressure,  the  liquid  is  forced  outward.     The 
simplest  way  of  finding  the  velocity  of  the  escaping  liquid  is  by 
an  application  of  the  principle  of  the  conservation  of  energy.    . 
A  small  mass  m  of  liquid  escaping  with  velocity  v  has  ^rav* 
units  of  kinetic  energy.     If  no  liquid  has  been  added  to  the  vessel 
during  the  escape  of  the  mass  m,  the  potential 
energy  of  the  liquid  in  the  vessel  must  have  dimin- 
ished by  an  amount  equal  to  £mt>2.     The  mass  m 
was  really  removed  from  the  part  of  the  liquid 
near  the  orifice,  but  the  change  of  the  state  of  the 
liquid  in  the  vessel  is  the  same  as  if  the  mass  m  had 
been  removed  from  the  surface;  and  the  change  of 
FIO.  102.         total  potential  energy  of  the  liquid  in  the  vessel  and 
of  the  escaping  liquid  is  the  same  as  if  a  mass  m 
had  been  lowered  from  the  surface  to  the  depth  of  the  orifice. 
Hence,  denoting  the  depth  of  the  orifice  below  the  surface  by 
h,  the  loss  of  potential  energy  is  mgh  and,  therefore, 


and  \m^ 

v = \/2gh 

Thus  the  velocity  of  escape  is  the  same  as  if  the  escaping  liquid 
had  fallen  freely  through  the  distance  of  the  orifice  below  the 
surface.  This  is  known  as  Torricelli's  Theorem.  It  was  first 
stated  by  a  pupil  of  Galileo  named  Torricelli,  who  also  discovered 
the  principle  of  the  barometer. 

Torriceili's  Theorem  may  also  be  deduced  from  Bernoulli's  theorem,  but 
we  shall  leave  the  deduction  as  an  exercise  for  the  reader. 

The  above  theorem  relates  only  to  the  velocity  of  the  par- 
ticles as  they  leave  the  orifice.    It  does  not  enable  us  at  once 
to  calculate  the  volume  that  escapes  in  a  given  time;  for  the 
cross-section  of  the  jet  contracts  for  a  short  distance  after 
leaving  the  vessel,  and  at  a  certain  point  reaches  a  minimum 
called  the  vena  contracta  (or  contracted  vein)  beyond  which  it      Fia.  103. 
expands.     If  the  area  a  of  the  cross-section  of  the  vena  con- 
tracta is  found,  the  volume  per  second  that  escapes  is  av.    The  ratio  of  a 
to  the  area  of  the  orifice  depends  on  the  velocity  of  escape  and  can  be 
changed  by  inserting  a  tube  (or  ajutage)  through  the  orifice. 


PROPERTIES  OF  FLUIDS 


137 


Fio.  104. 


204.  Pressure  Exerted  by  a  Stream. — When  a  stream  of  liquid 
meets  an  obstacle  and  is  arrested,  it  gives  up  its  momentum  to 
the  obstacle,  that  is,  it  exerts  a  force  on  the  obstacle. 
The  pressure  thus  produced  can  be  calculated  from 
the  velocity  of  the  water  and  the  amount  of  water 
that  impinges  per  second  on  this  obstacle.  On  this 
is  founded  a  method  of  measuring  the  velocity  of  a 
stream  (Pitot's  tube).  A  tube  bent  at  right  angles 

is  placed  in  the  stream  so  that  one  arm  points  hori- 
zontally up  stream  and  the  other  vertically  upward. 
If  the  water  were  at  rest,  the  liquid  would  rise  in 
the  vertical  arm  to  the  height  of  the  surface  of  the 
water,  but  the  pressure  of  the  stream  raises  it  higher 
and  from  this  additional  height  the  velocity  of  the 
stream  can  be  deduced. 


In  Ktot's  tube  the  case  of  $203  is  reversed.    Let  the  rise  of 


Fio.  105.— 
Pitot  tube  for 

ity  of  Stream?  level  be  h.     Suppose  the  liquid  to  be  continually  removed  at 
the  level  h.    In  any  time  the  total  decrease  of  kinetic  energy 
will    be  \md*  and  the  total  increase  of  potential  mgh.  f 

Hence  0a— 2gh  (approx.)  but  a  correction  factor  (slightly 
greater  than  unity)  is  necessary  because  the  tube  disturbs 
the  uniform  flow  of  the  stream. 


When  a  jet  impinges  on  an  obstacle  and  flows  off 
laterally,  the  pressure  exerted  is  that  due  to  the        FIO.  ioe. 
loss  of  the  momentum  of  the  liquid.     If  this  ob- 
stacle is  curved  so  that  the  motion  of  the  liquid  is 
reversed,  the  water  is  given  an  equal  momentum  in 
the  opposite  direction  and  the  force  exerted  on  the 
obstacle  is  doubled.    This  principle  is  taken  advantage 
of  in  the  construction  of  water-wheels. 

FIO.  lor.—  When  a  stream  strikes  an  obstacle  obliquely,  it  is 
A  disk  swept  partly  arrested  and  then  flows  down  along  the  sur- 
air  turns  pert  face  of  the  obstacle.  Thus  the  side  of  the  obstacle 
pendiouiarto  farther  up  stream  receives  more  momentum  than  the 

the  direction     .  .  .  .  .  , 

of  motion,  lower  side  and  so  tends  to  turn  more  nearly  perpen- 
dicular to  the  stream.  A  floating  log  free  only  to 
swing  about  its  middle  point  sets  itself  across  the  stream. 
A  leaf  falling  from  a  tree  tends  to  take  a  horizontal  position. 
The  effect  is  readily  illustrated  by  sweeping  through  the  air  a 


138        MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

square  disk  of  cardboard  which  is  connected  by  short  threads 
to  a  wire  frame  (Fig.  107). 

205.  The  Hydraulic  Ram. — Water  flowing  under  the  action  of 
gravity  tends  to  the  condition  in  which  it  would  be  in  equilibrium, 
and  in  which,  therefore,  all  parts  of  the  free  surface  would  be  at 
the  same  level.  This  is  the  meaning  of  the  statement  that 
"water  seeks  its  own  level."  Usually  it  is  only  by  the  means  of 
work  done  by  some  force  other  than  gravity  that  water  can  be 
i-aised  to  a  higher  level.  In  the  hydraulic  ram  a  small  fraction  of 

the  water  in  a  stream  is  raised  to  a  high 
level  by  a  self-acting  mechanism  which 
does  not  need  any  external  power. 

When  a  stream  of  water  in  a  pipe  is 
suddenly  stopped,  for  example  when  a 
faucet  is  turned  off,  the  momentum  of 
the  water,  which  may  be  very  large,  is 

Fia.  108.— Principle  of  ,    .  ,  ,.  -,   ,, 

hydraulic  ram.  stopped  in  a  very  short  time  and  there- 

fore the  force  exerted  by  the  water  on 

the  pipes  may  be  very  much  larger  than  that  which  the  water 
exerts  after  it  has  come  to  rest.  In  the  hydraulic  ram  this 
momentary  intense  pressure  is  used  to  drive  water  into  an  air- 
chamber  such  as  is  used  in  a  force-pump. 

Momentary  interruptions  of  the  current  are  caused  by  the  open- 
ing and  closing  of  a  valve  which  works  automatically  in  a  vertical 
direction.  The  weight  of  the  valve  is  such  that,  when  it  is  closed 
and  the  water  is  at  rest,  the  pressure  of  the  water  on  the  lower 
surface  of  the  valve  is  not  sufficient  to  keep  it  closed;  hence  it 
opens  and  allows  the  stream  to  start.  The  stream  when  in  mo- 
tion carries  the  valve  with  it,  again  closing  it  and  arresting  the 
motion. 

Some  of  the  potential  energy  of  the  head  of  water  is  transformed 
into  kinetic  energy  of  the  flowing  stream,  and  this  is  partly 
changed  into  potential  energy  of  the  compressed  air,  which  again 
is  changed  into  potential  energy  of  the  water  at  the  top  of  the 
delivery  tube.  Only  a  small  part  of  the  water  is  finally  raised  to 
a  higher  level  than  its  original  one,  and  its  gain  of  potential 
energy  is  compensated  by  the  loss  of  potential  energy  of  the 
remainder. 


PROPERTIES  OF  FLUIDS  139 

Molecular  Properties  of  Liquids 

206.  Molecular  Forces. — Between  the  particles  of  a  solid  or  of 
a  liquid  there  are  attractions  that  keep  the  body  together,  unless 
these  forces  are  overcome  by  external  forces.     To  show  directly 
the  existence  of  these  forces  between  the  particles  of  a  liquid  is 
very  difficult,  since  a  liquid  so  readily  changes  its  shape.     It  has, 
however,  been  found  possible  to  fill  a  glass  tube  with  water  at  a 
high  temperature  and  then  seal  the  tube;  the  water,  on  cooling, 
continued  to  fill  the  tube,  without  contracting,  until  it  exerted  a 
tensile  force  of  over  seventy  pounds  per  square  inch  upon  the 
walls  of  the  tube.     The  water  would,  in  such  an  experiment, 
stand  a  much  higher  stress,  if  it  were  possible  to  free  it  perfectly 
from  absorbed  gases.     It  is  this  attraction  between  the  particles 
of  a  liquid  that  has  to  be  overcome  when  a  liquid  is  evaporated; 
and,  from  the  heat  required  for  evaporation,  it  can  be  calculated 
that  the  attractions  between  the  particles  are  very  powerful  and 
produce  a  very  great   internal  pressure   across  any  imaginary 
plane  in  the  liquid. 

From  the  above  it  might  be  thought  that  a  body  immersed  in 
a  liquid  would  feel  the  effect  of  this  great  internal  pressure. 
That  such  is  not  the  case  is  due  to  the  fact  that  the  molecular 
forces  of  attraction  are  sensible  only  when  the  distances  between 
the  particles  are  exceedingly  small.  (§160)  Now  the  thinnest 
solid  that  it  is  possible  to  insert  in  a  liquid  separates  the  particles 
so  far  that  the  attractions  between  them  are  negligible,  and 
thus  the  pressure  on  an  immersed  solid  is  merely  that  due  to  the 
causes,  gravitation  and  pressure  on  the  boundary,  considered 
earlier. 

"The  distance  to  which  the  force  of  attraction  is  sensible  is 
called  the  range  of  molecular  forces.  Any  particle  of  a  liquid  is 
attracted  by  all  particles  that  lie  within  this  range  and  these  are 
contained  within  a  sphere.  This  sphere,  whose  radius  is  the 
range  of  molecular  forces,  may  be  called  the  sphere  of  influence  of 
a  particle. 

207.  Surface  Tension. — The  molecular  forces  of  which  we  have 
been  speaking  produce  certain  remarkable  effects  at  the  surface 
of  a  liquid.     The  surface  of  a  liquid  tends  to  contract  to  the  smallest 
area  admissible.     Thus  a  drop  of  water  falling  through  the  air 


140        MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

becomes  spherical,  since  the  sphere  is  the  figure  of  least  surface 
for  a  given  volume.  The  same  is  true  of  a  drop  of  liquid  lead 
falling  in  a  shot  tower;  the  drop  solidifies  during  the  fall  and  is 
found  to  be  spherical  when  the  fall  is  sufficient  to  allow  it  to 
become  perfectly  solid  while  in  the  air.  A  mixture  of  alcohol 
and  water  can  be  prepared  of  the  same  density  as  an  oil,  and  a 
large  drop  of  the  mixture  floating  totally  immersed 
in  the  oil  is  spherical.  When  the  end  of  a  stick  of 
sealing-wax  or  of  a  glass  rod  is  melted  in  a  flame, 
it  tends  to  the  spherical  form.  A  beautiful  illus- 
tration  of  the  tendency  of  a  liquid  surface  to  con- 
^rac^  consists  in  forming  a  film  from  a  soap- 
bubble  solution  on  a  ring  of  wire,  to  which  a  loop 
of  silk  has  been  loosely  attached  so  that  the  loop  floats  in  the 
film;  when  the  film  is  broken  inside  the  loop  the  latter  becomes 
circular.  In  shrinking  to  the  form  of  least  area  the  film  pulls 
the  loop  into  the  form  of  greatest  area  for  a  given  periphery,  and 
this  is  a  circle. 

208.  Explanation  of  Surface  Tension. — Consider  the  condition 
of  a  particle  at  A  in  the  body  of  a  liquid,  and  that  of  a  particle  at 
B,  at  less  than  the  range  of  molecular  forces  from  the  surface. 
The  particle  at  A  is  equally  attracted  on  all  sides  by  the  particles 
around  it,  but  the  particle  at  B  is  more  attracted  inward  than 
outward,  since  a  sphere  with  center  at  B  and 
the  range  of  molecular  forces  as  radins  lies 
partly  outside  of  the  liquid.  To  take  a 
particle  from  A  to  B,  work  must  be  done 
against  this  inward  attraction.  Fjo  no 

Now,  when  the  surface  of  a  liquid  is 
increased,  for  example  when  a  soap  film  is  stretched,  more  par- 
ticles are  drawn  into  the  surface;  hence  some  work  is  done  by 
the  stretching  force  and  therefore  an  opposing  force  is  over- 
come. But  the  stretching  force  required  is  parallel  to  the  sur- 
face; hence  the  liquid  exerts  an  opposing  or  contractile  force 
parallel  to  the  plane  of  the  surface,  and  this  force  is  what  we 
call  the  surface  tension.  Thus  we  explain  the  existence  of  a 
tension  in  the  surface  of  a  liquid  by  showing  that  it  is  in  accord- 
ance with  the  principle  of  work.  At  present  our  knowledge  of 
the  state  of  the  particles  near  the  surface  is  too  imperfect  to 


PROPERTIES  OF  FLUIDS 


141 


FIG.  in.— 

Surface 

tension  i& 

the  force 

across  unit 

length. 


enable  us  to  describe  their  condition  more  precisely  and  to  show 
how  the  state  of  tension  along  the  surface  is  produced. 

If  a  line  be  imagined  drawn  along  the  surface  of  a  liquid,  the 
part  of  the  surface  on  one  side  of  the  line  pulls  on  the 
part  on  the  other  side,  and  if  the  length  of  the  line  be 
supposed  one  cm.  the  pull  in  dynes  is  taken  as  the 
magnitude  of  the  surface  tension,  T,  of  the  liquid. 
The  measure  of  surface  tension  is  the  force  of  contrac- 
tion across  a  line  of  unit  length  in  the  surface. 

209.  Methods  of  Measuring  Surface  Tension.  —  Sur- 
face tension  manifests  itself  in  many  ways  and,  as 
almost  any  of  its  effects  may  be  made  the  basis  of  a 
method  of  measuring  it,  the  methods  that  have  been  employed 
are  numerous.     When  the  liquid  can  be  formed  into  a  thin  sheet, 
as  in  the  case  of  a  soap  solution,  a  direct  method  of  measuring  it 

may  be  used;  a  film  may  be  formed  on  a  wire  frame 
of  which  one  side  is  movable;  if  the  force  required  to 
hold  this  side  at  rest  against  the  surface  tension  is  F, 
and  the  length  of  the  movable  side  is  I,  the  tension  in 
each  surface  of  the  film  is  F/21. 

To  draw  a  horizontal  wire  up  through  the  surface  of 
a  liquid  the  tension  of  the  surface  must  be  overcome, 
and  from  the  force  required  the  surface  tension  may 
be  calculated. 

The  movement  of  minute  waves  or  ripples  on  the  surface  of  a 
liquid  is  due  chiefly  to  the  surface  tension  of  the  liquid,  and  from 
the  wave-lengths  of  the  ripples  and  their  veloci- 
ties  we  can   find  the  magnitude  of  the  surface 
tension. 

The  rise  of  liquid  in  a  capillary  tube  depends, 
as  we  shall  see  later,  on  the  surface  tension  of  the 
liquid,  and  this  affords  another  method  of  measure- 
ment. 

210.  Contact  of  Liquid  and  Solid.—  The  general 
free  surface  of  a  liquid  is  horizontal;  but,  where 

the  liquid  is  in  contact  with  a  solid,  the  surface  is  usually  curved, 
the  direction  and  amount  of  the  curvature  being  different  for 
different  liquids  and  different  solids.  Water  in  contact  with 
a  vertical  surface  of  glass  is  curved  upward,  and  mercury  in 


I 
F 

FIG.  112.— 

Stretching 

a  film. 


Fl°-    us.— 


142        MECHANICS  AND  THE  PROPERTIES  OF  MATTER 


Fio.  114.— 
Contact  of  mer- 
cury and  glass. 


the  same  circumstances  is  curved  downward.      These,  for   a 
reason  stated  later,  are  called  capillary  phenomena. 

The  contact  angle  of  the  wedge-shaped  part  of  the  liquid 
between  the  free  surface  of  the  liquid  and  the  surface  of  the  solid 
is  called  the  angle  of  capillarity.     The  size  of  the  angle  in  any 
case  depends  on  the  purity  of  the  liquid  and  the  cleanness 
of  the  solid  surface.     Thus  for  very  pure  water  in 
contact  with  clean  glass  the  angle  is  0°;  but  with 
slight  contamination,  even  such  as  is  caused  by 
exposure  to  air,  the  angle  may  become  as  large  as 
25°  or  more.     For  perfectly  pure  mercury  and  glass 
the  angle  is  about  148°,  but  slight  contamination 
reduces  it  to  140°  or  less;  for  turpentine  it  is  17°, 
for  petroleum  26°  and  so  on. 

211.  Level  of  Liquids  in  Capillary  Tubes. — When  a  glass  tube 
of  very  fine  bore  (or  so-called  capillary  tube) ,  open  at  both  ends, 
is  placed  vertically  with  its  lower  end  in  a  vessel  of  liquid,  the 
surface  of  the  liquid  in  the  tube  is  usually  higher  or  lower  than  the 
general  level  of  the  surface  in  the  vessel.  When  the  liquid  is 
water  or  alcohol  the  surface  is  elevated  in  the  tube;  when  the 
liquid  is  mercury  the  surface  is  depressed.  For  a  given  liquid  the 
amount  of  elevation  or  depression  is  greater  the  smaller  the  bore 
of  the  tube,  being,  in  fact,  inversely  as  the 
diameter  of  the  bore.  For  tubes  of  other 
materials  than  glass  similar  effects,  de- 
pending in  amount  on  the  material  of  the 
tube,  are  observed. 

There  are  similar  elevations  and  depres- 
sions between  two  glass  plates  standing 
close  together  in  a  liquid.  These  eleva- 
tions and  depressions  and  the  curvature 
of  a  liquid  surface  in  contact  with  a  solid 
are  usually  grouped  under  the  general  title  of  Capillarity. 

Assuming  the  existence  of  the  invariable  angle  of  capillarity 
at  which  a  liquid  meets  a  solid,  we  can  give  a  simple  explanation 
of  capillary  elevations  and  depressions. 

Consider  the  case  when  the  liquid  is  elevated.  The  liquid  in 
the  tube  meets  the  tube  in  a  circle  of  radius  r  equal  to  the  radius 
of  the  bore,  and  at  every  point  of  the  circle  the  angle  of  contact 


FIQ.  115. — Water  in  capillary 
tubes. 


PROPERTIES  OF  FLUIDS  143 

is  the  angle  of  capillarity  a.     Thus  the  surface  tension  of  the 
liquid  pulls  on  the  tube  in  the  direction  PQ  inclined  at  a  to  the 
length  of  the  tube;  and  the  tube  therefore  reacts 
with  an  equal  pull  in  the  direction  QP.     The 
amount  of  the  pull  per  unit  length  of  the  cir- 
cumference of  the  circle  of  contact  is  T,  and  the 
component  of  this,  parallel  to  the  length  of  the 
tube,  is  Tcosa.     For  the  whole  circumference 
of  the  circle  of  contact  the  sum  of  these  com- 
ponents is  2nrT  cos  a.     This  is  an  upward  force 
on  the  liquid  in  the  tube,  and  it  draws  the  liquid     == 
upward  until  the  weight  of  the  liquid  elevated 
above  the  ordinary  surface  equals  the  support-          FIO.  ne. 
ing    force.      If    the    mean  elevation  is  h,   the 
volume  of  the  supported  column  is  rcr2h  and  its  weight  nr*hpg  in 
dynes.     Hence 

""  cos  a 


gpr 

Thus  the  elevation  is  directly  as  the  surface  tension  and  inversely 
as  the  radius  of  the  tube.  By  measuring  the  elevation  and  the 
radius  and  finding  a  by  some  other  method,  the  value  of  T  for  any 
liquid  may  be  obtained. 

212.  Elevation  between  Plates. — The  above  method  of  proof  may  also 
be  extended  to  the  case  of  a  liquid  between  parallel  plates  (Fig.  116).     In 
this  case  the  surface  of  the  liquid  meets  the  surfaces  of  the  plates  in  straight 
lines.     Let  the  distance  between  the  plates  be  d.     Consider  the  equilibrium 
of  the  liquid  contained  between  the  plates  and  two  vertical  planes  perpen- 
dicular to  the  plates  and  at  unit  distance  apart.     The  pull  of  the  surface 
tension  at  the  top  is  2T  cos  a  and  the  weight  of  the  liquid  supported  is 
dhpg.     Hence 

2T  cos  a 
9pd 

Thus  the  elevation  is  the  same  for  two  parallel  plates  as  for  a  tube,  if  the 
distance  between  the  plates  equals  the  radius  of  the  tube. 

213.  Pressure  Caused  by  a  Curved  Surface  under  Tension. — 

Since  the  liquid  in  a  capillary  tube  is  elevated  above  or  depressed 
below  the  ordinary  level,  the  pressure  beneath  the  curved  surface 
must  be  less  or  greater  than  the  pressure  at  the  general  surface. 


144         MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

When  the  effect  is  a  depression  (mercury  in  glass) ,  the  depressed 
surface  is  curved  downward  and  the  tension  in  the  surface  pro- 
duces a  pressure,  just  as  the  tension  in  a  rubber  sheet  stretched 
over  a  ball  produces  pressure  on  the  ball.  When  the  effect  is  an 
elevation,  the  stretch  on  the  upward  curved  surface  tends  to 
draw  the  liquid  in  the  surface  layer  away  from  the  liquid  below 
and  so  produces  a  state  of  tension  or  diminution  of  pressure  be- 
neath the  surface.  From  the  amount  of  the  elevation  or  depres- 
sion we  can  calculate  the  change  of  pressure  thus 
caused.  In  the  case  of  an  elevation  to  a  height  h  the 
pressure  must  be  less  than  the  pressure  at  the  ordi- 
nary level,  which  is  atmospheric  pressure,  by  gph  or, 
(§211),  2Tcosa/r.  Here  r  is  the  radius  of  the 
tube.  If  we  denote  the  radius  of  the  spherical  sur- 
face by  R,  R  cos  a  =  r.  Hence  the  pressure  beneath 
FIO.  117.  the  concave  surface  is  less  than  that  of  the  at- 
mosphere above  by  2T/R.  The  same  applies  to  the 
pressure  produced  on  the  concave  side  of  a  depressed  surface. 
This  difference  of  pressure  on  the  two  sides  is  due  entirely  to 
the  tension  and  the  curvature  of  the  surface. 

In  the  case  of  a  spherical  soap-bubble  there  are  two  surface 
tensions  to  be  considered,  one  on  the  inner  side  of  the  film  and  the 
other  on  the  outer  side.  Hence  the  excess  pressure  inside  the 
bubble,  due  to  the  tension  and  curvature  of  the  film,  is  4T/R. 

A  cylindrical  surface  in  a  state  of  tension  also  produces  pressure  on  the 
concave  side.  This  is  deduced,  as  above,  from  the  elevation  or  depression 
of  a  liquid  of  surface  tension  T  between  two  parallel  plates  at  a  distance  d 
apart.  If  R  is  the  radius  of  the  cylindrical  surface  of  the  liquid  R  cos  a  •=  ±d. 
Hence  (§211)  p^T/Rt  and  this  is  therefore  the  pressure  on  the  concave 
side  due  to  the  tension  T  in  a  cylindrical  surface  of  radius  R.  In  the  case 
of  a  cylindrical  soap-bubble  of  radius  R  the  tension  in  each  surface  pro- 
duces pressure  T/R.  Hence  the  pressure  inside  is  greater  than  that  outside, 
by  2T/R 

214.  Other  Effects  of  Surface  Tension.— When  the 

angle  of  capillarity  of  a  liquid  in  contact  with  a  solid    FIQ.  us. — Water  be- 
is  small,  the  liquid,  in  its  attempt  to  establish  this        tween  glass  plates, 
small  angle,  spreads  out  on  the  surface  of  the  solid; 
that  is,  the  liquid  is  one  that  wets  the  solid.     Thus  a  drop  of  water  let  fall 
on  clean  glass  spreads  out,  the  angle  of  capillarity  being  small.     A  drop  of 
mercury  on  a  glass  plate  has  no  tendency  to  spread  but  gathers  into  a  ball. 

A  film  of  water  between  two  glass  plates  makes  it  difficult  to  draw  the 


PROPERTIES  OF  FLUIDS 


145 


plates  apart  by  a  force  normal  to  their  surfaces.  The  liquid  tends  to  spread 
over  both  plates  and  becomes  concave  outwards,  so  that  the  pressure  within 
it  is  less  than  the  atmospheric  pressure  which  acts  on  the  outside  of  the 
plates,  and  this  produces  an  apparent  attraction  between  the  plates. 

When  an  attempt  is  made  to  blow  out  a  glass  tube  containing  numerous 
detached  drops  a  surprising  resistance  is  experienced.  Each  drop  becomes 
concave  on  the  side  of  high  pressure  and  the  total  resistance  is  the  sum 
of  the  pressures  exerted  by  these  concave  surfaces. 

Small  bodies,  such  as  straws  and  sticks,  floating  on  the  surface  of  a 
liquid  usually  attract  and  gather  into  groups.  Let  us  represent  two  such 
bodies  by  small  vertical  plates.  If  the  liquid  wets  both  it  rises  between 
them,  and  the  pressure  in  the  elevated  portion  is  less  than  the  atmospheric 
pressure  on  the  outer  side  of  the  plates.  Hence  the  plates  are  pushed 
together.  If  the  liquid  does  not  wet  either  plate  it  is  depressed  between 
them;  the  pressure  above  the  depressed  part  is  atmospheric,  while  the 
pressure  in  the  liquid  on  the  outer  sides  of  the  plates  is  greater  than  atmos- 
pheric and  the  plates  are  pushed  together.  If  the  liquid  wets  one  plate 


Fia.  119. — Capillary  attractions  and  repulsions. 

but  not  the  other  there  is  a  part  of  each  plate  on  which  the  pressure  on 
the  inside  is  greater  than  that  on  the  outside;  hence  an  apparent  repulsion 
results.  (Balls  of  paraffine  wax  some  of  which  are  lamp-blacked,  floating 
on  water,  will  illustrate  all  three  cases.) 

Any  dissolved  substance  or  impurity  changes  the  surface  tension  of 
water.  This  explains  the  irregular  motions  of  small  particles  of  camphor 
dropped  on  clean  water.  At  some  points  the  camphor  dissolves  more  rapidly 
than  at  other  points,  and  near  the  former  the  surface  tension  of  the  water 
is  weakened  so  that  the  pull  on  the  opposite  side,  where  the  tension  is 
greater,  prevails  and  causes  irregular  motion. 

216.  Diffusion  of  Liquids. — The  gradual  mixture  of  two  liquids 
which  come  into  contact  is  called  diffusion.  It  takes  place  on  a 
large  scale  where  fresh  water  from  a  river  flows  out  into  the 
ocean.  It  may  be  illustrated  on  a  small  scale  by  pouring  a  solu- 
tion of  a  colored  salt  into  a  tall  vessel  and  then  cautiously  cover- 
ing the  colored  solution  with  a  layer  of  water.  The  particles  of 
each  liquid  are  in  motion  and  begin  to  make  their  way  across  the 
interface  and,  after  a  long  time,  the  whole  vessel  is  filled  with  a 
mixture  of  the  same  constitution  throughout.  Stirring  has  the 

10 


146        MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

effect  of  increasing  the  area  of  contact  of  the  liquids  and  so  pro- 
motes diffusion.  But  some  substances  such  as  oil  and  water  will 
not  diffuse  into  one  another,  or  "mix"  probably  because  of 
their  very  different  internal  attractions. 

Let  us  denote  the  two  diffusing  liquids  by  A  and  B,  and  let  us 
suppose  that  initially  A  occupies  the  lower  half  of  a  tall  jar  and 
B  the  upper  half.  The  concentration  of  either  of  the  liquids  at 
any  point  is  its  mass  per  unit  volume  at  that  point  (i.e.,  its  den- 
sity at  the  point  if  the  other  liquid  be  imagined  absent  without 
the  first  being  disturbed).  The  liquid  A  diffuses  vertically  up- 
ward, that  is,  from  places  of  high  concentration  to  places  of  low 
concentration.  The  gradient  of  concentration  in  any  direction  is 
the  rate  at  which  the  concentration  falls  off  in  that  direction;  if 
the  rate  of  fall  per  unit  of  distance  is  unity,  the  gradient  of  con- 
centration is  unity.  The  general  law  of  diffusion  is  that  the 
rate  of  diffusion  for  each  liquid  is  proportional  to  the  gradient  of 
concentration  of  that  liquid.  The  coefficient  of  diffusion,  or  the 
diffusivity  of  the  liquid,  is  the  mass  in  grams  that  crosses  unit  area 
in  a  day  when  the  gradient  of  concentration  is  unity.  This  con- 
stant can  be  found  from  observations  of  the  density  at  various 
points  along  the  direction  of  diffusion,  made  by  means  of  beads 
of  different  densities  floating  in  the  liquid,  and  in  various  other 
ways.  The  following  table  contains  the  coefficients  of  diffusion 
of  various  substances  into  water  at  the  temperature  (Centigrade) 
stated. 

Hydrochloric  acid 1 . 74  at    5° 

Common  salt 0 . 76  at    5° 

Common  salt 0.91  at  10° 

Sugar 0.31  at    9° 

Albumen 0.06  at  13° 

Caramel 0.05  at  10° 

From  the  above  it  will  be  seen  that  liquids  vary  widely  in  dif- 
fusivities.  Substances  of  high  diffusivity  are  called  crystalloids 
and  those  of  low  diffusivity  are  called  colloids.  The  former 
group  includes  mineral  acids,  salts  and  substances  generally  that 
form  crystals  (whence  the  name),  while  the  latter  includes  gums, 
albumens,  starch,  and  glue  (the  name  being  derived  from  the 
Greek  for  glue).  Crystalloids  dissolved  in  water  produce  many 
marked  changes  in  its  properties;  colloids  in  water  form  jellies, 


PROPERTIES  OF  FLUIDS 


147 


which  seem  to  consist  of  a  semi-solid  framework  holding  the  liquid 
in  its  meshes.  Colloids  have  large  and  complex  molecules  and  it 
is,  perhaps,  to  this  fact  and  to  the  consequent  slower  motions  of 
the  molecules  that  their  small  diffusivities  are  due.  They  are 
comparatively  tasteless,  as  they  do  not  diffuse  and  reach  the 
nerve  terminals.  Their  low  rates  of  diffusion  also  render  them 
indigestible.  Through  a  layer  of  a  colloidal  jelly  crystalloids 
will  diffuse  almost  as  rapidly  as  through  water,  but  colloids  not 
at  all. 

216.  Diffusion  through  Membranes.  Osmosis. — Through  cer- 
tain membranes  which  have  no  visible  pores,  many  liquids  will 
diffuse  readily.  Thus  through  a  partition  of  rubber  between 
water  and  alcohol  the  alcohol  will  pass  rapidly,  while  the  passage 
of  the  water  is  barred.  If  animal  membranes  are  wet  by  water, 
it  readily  passes  through.  A  method  of  separating  crystalloids 
and  colloids,  called  dialysis,  depends  on  the  different  rates  at 
which  these  substances  pass  through  such  a  membrane  as  parch- 
ment paper.  The  diffusion  of  substances  through  such  septa  is 
called  osmosis. 

Some  membranes  allow  one  constituent  of  a  mixed  liquid  or 
solution  to  pass,  while  barring  the  other  constituent;  such  mem- 
branes are  called  semi-permeable.  One  such  is  ferrocyanide  of 
copper,  formed  in  the  pores  of  a  porous  partition  by  the  reaction 
between  ferrocyanide  of  potassium  on  one  side  and 
copper  sulphate  on  the  other.  When  such  a  mem- 
brane separates  water  and  the  aqueous  solution  of  any 
one  of  various  salts,  the  salt  does  not  pass,  but  the 
water  passes  in  both  directions,  though  more  rapidly 
toward  the  solution  than  in  the  opposite  direction. 
If  the  solution  be  in  a  tube  the  lower  end  of  which, 
closed  by  a  plug  of  the  membrane,  is  dipped  in  water, 
the  level  in  the  tube  will  rise  until  (provided  the  mem- 
brane does  not  break)  the  column  is  of  such  a  height 
that  its  pressure  prevents  further  flow.  This  pressure 
is  called  the  osmotic  presssure  of  the  solution.  Its  mag- 
nitude, for  very  weak  solutions,  is  proportional  to  the  concentra- 
tion, that  is,  to  the  number  of  molecules  of  the  dissolved  salt  per 
unit  volume.  For  a  large  number  of  salts  the  pressure  is  the  same 
for  solutions  that  contain  the  same  number  of  molecules  of  the 


Fia.  120.— 
Osmotic 
pressure. 


148        MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

salt  in  unit  volume.  For  various  other  salts  the  osmotic  pressure 
for  a  given  number  of  molecules  per  unit  volume  is  two  (or  some 
whole  number  of)  times  greater  than  for  the  first  group;  this  is 
possibly  due  to  the  molecules  being  resolved  into  atoms  in  the 
solution,  the  atoms  acting  independently.  But  the  full  explana- 
tion of  osmosis  and  osmotic  pressure  is  a  matter  of  much  dispute. 
One  remarkable  fact  may  be  noted,  namely,  that  the  osmotic 
pressure  for  a  given  number  of  molecules  (or  of  dissociated  atoms) 
in  an  aqueous  solution  is  equal  to  the  pressure  that  these  mol- 
ecules (or  atoms)  would  produce  if  freely  flying  as  gaseous  par- 
ticles in  the  space  occupied  by  the  solution.  It  is  also  note- 
worthy that  the  osmotic  pressure  increases  in  the  same  way  and 
at  the  same  rate  with  rise  of  temperature  as  the  pressure  of  a 
gas  does. 

Osmosis  plays  an  important  part  in  many  processes  that  take 
place  in  the  bodies  of  animals  and  plants. 

PROPERTIES  OF  GASES 

217.  A  gas  has  already  been  defined  as  a  fluid  which  has  no 
definite  volume  of  its  own  independent  of  the  containing  vessel, 
but  expands  so  as  to  occupy  any  vessel  in  which  it  is  contained. 
Gases  have  the  same  properties  as  liquids  in  all  respects  which 
depend  on  the  fact  that  the  shear  modulus  of  a  fluid  is  zero.     The 
pressure  at  a  point  in  a  gas  is  the  same  in  all  directions  (§184). 
The  pressure  of  a  gas  on  a  surface  is  normal  to  the  surface  (§183). 
Pressure  applied  to  any  part  of  the  boundary  is  equally  trans- 
mitted in  all  directions  (Pascal's  Principle  §189).     A  body  im- 
mersed in  a  gas  is  buoyed  up  with  a  force  equal  to  the  weight  of 
the  gas  displaced  (Archimedes'  Principle  §191).     The  pressure 
in  a  gas  increases  with  its  depth  at  a  rate  expressed  by  gph,  as  in 
the  case  of  liquids  (§185).     Gases  also  show  the   property  of 
internal  friction  or  viscosity,  and  the  definition  of  the  coefficient 
of  viscosity  of  a  gas  is  the  same  as  that  of  a  liquid.     Some  of 
these  properties  are  of  special  importance  in  the  case  of  a  gas 
and  call  for  separate  treatment. 

218.  Pressure  of  the  Atmosphere. — A  very  important  example 
of  the  pressure  of  a  gas  is  the  pressure  exerted  by  the  earth's 
atmosphere.     The  atmosphere,  consisting  chiefly  of  oxygen  and 


PROPERTIES  OF  GASES  149 

nitrogen,  is  held  to  the  earth  by  the  gravitational  attraction 
between  it  and  the  earth.  The  total  pressure  on  the  surface  of 
the  earth  is  the  total  attraction  between  the  earth  and  the 
atmosphere,  that  is,  the  weight  of  the  atmosphere.  The  pressure 
on  any  horizontal  area  of  the  earth's  surface  is  the  weight  of  all 
the  air  vertically  above  that  area.  At  the  top  of  a  mountain  the 
pressure  is  less  than  at  sea  level,  since  less  of  the  atmosphere  is 
above. 

Galileo  discovered  that  air  had  weight  by  weighing  a  glass 
globe  containing  air  and  then  re-weighing  it  when  he  had  forced 
more  air  into  it.  His  friend  and  pupil  Torricelli 
found  (in  1643)  that,  when  a  tube  33  inches  long 
filled  with  mercury  and  closed  at  one  end  was  in- 
verted in  a  dish  of  mercury,  the  mercury  stood  at  a 
height  of  about  30  inches  in  the  tube,  thus  leaving  a 
vacuum  above.  This  is  known  as  Torricelli's  Experi- 
ment. He  thus  disproved  the  previous  view  that 
"  Nature  abhors  a  vacuum*,"  and  was  led  to  infer  that 
the  pressure  of  the  atmosphere  on  any  area  equals 
that  of  a  column  of  mercury  about  30  inches  high 
and  of  a  cross-section  equal  to  the  area.  On  hearing 
of  Torricelli's  experiment,  Pascal  reasoned  that  the  FIG.  121 
pressure  should  be  less  and  the  column  of  mercury  in 
Torricelli's  tube  lower  at  the  top  of  a  mountain  and  he 
wrote  to  a  relative,  who  lived  near  the  Puy  de  Dome  in  Auvergne, 
to  make  the  test.  The  result  confirmed  his  conjecture. 

219.  The  Mercurial  Barometer. — Torricelli's  tube  was  the  first 
and  simplest  barometer  or  pressure-gauge  for  measurement  of  the 
pressure  of  the  atmosphere.  The  most  accurate  mercurial  ba- 
rometer of  the  present  day  is  a  Torricellian  tube  with  a  scale  and 
vernier  for  accurate  measurement  of  the  height  of  the  mercury 
column,  and  a  device  by  which  the  mercury  in  the  cistern  may 
be  readily  brought  to  a  definite  height.  In  Fortin's  cistern 
barometer  (Fig.  122)  the  cistern,  C,  has  a  leather  bottom,  S,  the  cen- 
ter of  which  rests  on  a  screw,  V.  By  turning  the  screw  the  level 
of  the  mercury  in  the  cistern  can  be  raised  or  lowered  so  that 
when  the  barometer  is  read  the  level  of  the  mercury  in  the  cistern 
shall  always  be  the  same,  namely,  zero  of  the  scale  on  which  the 
height  of  the  barometer  is  read.  Without  such  an  adjustment. 


150        MECHANICS  AND  THE  PROPERTIES  OF  MATTER 


the  level  of  the  mercury  in  the  cistern  would  fall  or  rise  as  the 
height  of  the  mercury  in  the  tube,  T,  rose  or  fell.  That  the  level 
of  the  mercury  in  the  cistern  may  be  observed, 
the  upper  part  of  the  cistern  is  of  glass  and  a 
small  ivory  stud,  0,  projecting  downward  from 
the  top  of  the  cistern,  is  adjusted  by  the  maker, 
so  that  its  end  is  on  a  level  with  the  zero  of  the 
scale.  The  image  of  the  stud  in  the  surface  of 
the  mercury  is  observed  and  when,  as  the  level 
of  the  mercury  is  raised  by  the  screw,  the  end 
of  the  stud  and  the  end  of  its  image  just  meet, 
the  surface  of  the  mercury  is  at  the  zero  of  the 
scale.  In  filling  such  a  barometer  care  must 
be  taken  that  no  air  remains  in  the  mercury, 
and,  for  this  purpose,  after  the  tube  ^ 
has  been  filled  it  is  inverted  and 
the  mercury  boiled  so  that  the  air 
is  expelled.  The  mercury  in  the 

,         cistern  becomes  somewhat  tarnished 

H^  in  course  of  time  and  the  image  of 

the  stud  ceases  to  be  distinct. 

A  simpler  form  of  barometer  is 
Bunsen's  siphon  barometer.  In  this 
there  is  no  cistern,  but  the  lower  end  of  the  tube  is 
turned  vertically  upward.  The  difference  of  level 
in  the  open  and  in  the  closed  end  is  the  barometric 
height.  Thus  readings  of  both  ends  of  the  mercury 
column  are  necessary.  Scales  are  etched  on  both 
branches;  the  one  on  the  longer  arm  reads  upward 
and  that  on  the  shorter  arm  reads  downward.  The 
two  scales  are  usually  laid  off  with  the  same  position 
for  the  zero,  so  that  the  sum  of  the  two  readings  is 
the  height  of  the  barometer. 

Another  form  of  barometer  is  the  Aneroid  (Greek 
aneros  =  dry)  barometer  in  which  no  liquid  is  used. 
It  consists  of  a  metallic  box  exhausted  of  air,  with 
a  thin  metallic  cover.  Changes  in  atmospheric 
pressure  cause  slight  changes  of  curvature  in  the  cover,  and  by 
means  of  a  multiplying  system  of  levers  these  changes  are 


Fio.  122. — Cistern  of 
Fortin'g   barometer. 


Fio.  123.— 

Bunsen's 

siphon 

barometer. 


PROPERTIES  OF  GASES  151 

transmitted  to  a  pointer,  which  moves  around  a  circular  scale 
that  is  graduated  in  cms.  or  inches  so  as  to  correspond  to  the 
readings  of  the  mercurial  barometer.  This  form  of  barometer 
is  more  convenient  for  travellers,  but  it  has  the  disadvantage 
that  its  index  must  frequently  be  reset  by  comparison  with  the 
mercury  barometer. 

220.  Uses   of    the    Barometer. — A   knowledge  of  barometric 
pressure  is  of  great  importance  in  weather  forecasting.     The 
governments  of  the  United  States  and  other  civilized  nations 
maintain  a  large  number  of  stations  where  records  of  the  baro- 
meter are  kept.    From  simultaneous  readings  over  a  wide  area 
the  direction  in  which  storms  (or  areas  of  low  pressure)  will  move 
can  be  predicted.     Such  predictions  lead  annually  to  the  saving 
of  thousands  of  lives,  and  of  much  valuable  property  in  shipping. 

Since  the  atmospheric  pressure  is  less  at  higher  levels,  it  is 
possible  to  ascertain  the  height  of  a  mountain  by  observing  the 
atmospheric  pressure  at  the  top  and  at  the  bottom.  Near  sea- 
level  the  height  of  the  barometer  diminishes  by  about  0.1  inch 
for  every  80  feet  of  ascent;  but  as  the  elevation 
increases  the  rate  of  fall  diminishes  owing  to  the 
greater  rarity  of  the  air.  Allowance  must  be  made 
for  any  difference  of  temperature  at  the  two  stations 
of  observation. 

221.  Pressure  and  Volume  of  a  Mass  of  Gas. — Com- 
mon observation  shows  that  added  pressure  on  a  mass 
of  gas  diminishes  its  volume.      Thus,  in  pumping  up 

a  bicycle  tire,  a  large  volume  of  air  from  the  atmo-  £ 
sphere  is  forced  by  high  pressure  into  the  small  volume  (I 
of  the  tire.  Conversely,  diminution  of  pressure  allows  Fjo  4  _ 
a  gas  to  expand.  Against  the  pressure  exerted  on  a  Boyle's  tube 
gas  it  exerts  an  equal  and  opposite  pressure,  so  that  ^^^ 
it  is  immaterial  whether  we  speak  of  the  pressure  on  atmospheric, 
or  pressure  of  a  gas. 

The  law  connecting  the  volume  and  the  pressure  of  a  gas  is 
extremely  simple,  but  it  was  not  discovered  until  1662,  the  dis- 
coverer being  Robert  Boyle.  (Fourteen  years  later  Mariotte  re- 
discovered the  same  law.)  The  volume  of  a  gas  at  constant  tem- 
perature varies  inversely  as  its  pressure,  or,  denoting  the  pressure 
and  volume  by  P  and  V  respectively,  P V  =  a  constant.  Boyle 


152        MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

discovered  this  law  by  experiments  conducted  with  a  tube  bent 
as  in  Fig.  124,  the  shorter  arm  being  closed  and  containing  air 
and  mercury,  while  the. longer  was  open  and  was  filled  to  varying 
depths  with  mercury.  If  to  the  difference  of  level  in  the  two 
arms  the  height  of  the  mercury  barometer  at  the  time  is  added, 
the  sum  is  proportional  to  the  pressure  on  the  air,  while  the 
length  of  the  tube  occupied  by  air  is  proportional  to 
the  volume  of  the  air.  Thus  he  discovered  the  truth 
of  the  law  for  pressures  exceeding  an  atmosphere. 
For  pressures  below  an  atmosphere  he  used  a  straight 
tube  containing,  initially,  air  and  mercury  and  closed 
at  one  end;  the  open  end  was  then  plunged  into  a 
deep  vessel  of  mercury.  By  drawing  the  tube  to 
different  heights  the  volume  of  the  air  increased  with 
diminishing  pressure.  Thus  Boyle  verified  the  law 
for  pressure  less  than  an  atmosphere. 

222.  Deviations  from  Boyle's  Law. — While  the  law 
stated  by  Boyle  is  accurate  enough  for  all  ordinary 
for  pressure  practical  purposes,  careful  tests  have  shown  that  it  is 
atmospheric!  not  perfectly  accurate.  The  most  complete  tests 
were  made  by  Amagat.  He  found  that  in  the  case 
of  air,  while  the  pressure  is  being  increased  from  one  atmos- 
phere to  about  78  atmospheres,  PV  steadily  diminishes,  until  its 
value  is  0.98  of  its  value  at  one  atmosphere.  Thereafter,  with 
increasing  pressure,  PV  increases  so  that  at  3000  atmospheres  it 
has  a  value  4.2  times  its  initial  value.  In  the  first  stage  (that  is, 
up  to  78  atmospheres)  V  decreases  more  rapidly  than  Boyle's 
law  would  indicate;  thereafter  it  decreases  less  rapidly,  so  that  at 
3000  atmospheres  its  volume  is  4.2  times  what  it  would  be  if 
Boyle's  law  were  perfectly  accurate.  (It  may  be  noted  that  at 
3000  atmospheres  air  has  a  density  of  0.93,  nearly  equal  to  that 
of  water;  while  the  density  of  liquid  oxygen  at  its  critical  pressure 
is  about  0.7  and  that  of  liquid  nitrogen  about  0.4.) 

Other  gases,  show  similar  deviations  from  Boyle's  law;  but 
the  pressure  at  which  PV  is  a  minimum  is  widely  different  for 
different  gases,  and  so,  too,  is  the  magnitude  of  this  minimum 
value  of  PV. 

In  his  earlier  experiments  (1881)  Amagat  measured  pressures  by  a  very 
tall  manometer  in  a  mine  shaft.  Later  he  designed  a  special  gauge  for 


PROPERTIES  OF  GASES  153 

very  high  pressures.  This  consisted  of  two  opposed  pistons  of  very  dif- 
ferent diameters  in  separate  cylinders.  The  high  pressure,  Pt  was  applied 
to  the  small  piston,  of  area  a,  and  was  counterbalanced  by  a  much  smaller 
pressure,  p,  applied  to  the  large  piston,  of  area  A.  Evidently  Pa°*pA, 
and,  p  being  measured  by  a  mercury  manometer,  P  Was  deduced.  Very 
viscous  liquids  were  used  in  the  cylinders  to  diminish  leakage. 

Starting  with  the  view  that  a  gas  consists  of  flying  particles  the  impact 
of  which  produces  the  pressure  observed  in  a  gas,  Van  der  Waals  deduced 
the  following  formula  which  agrees  very  well  with  the  results  of  Amagat's 
experiments. 

(P+o/F2)(F-6)=a  constant, 

at  constant  temperature,  a  and  6  being  constants  that  are  different  for 
different  gases. 

223.  Modulus  of  Elasticity  of  a  Gas.  —  The  shear  modulus  of  a 
gas  being  zero,  a  gas  has  only  one  modulus,  namely  the  bulk 
modulus,  and  this  is  (when  the  gas  is  kept  at  constant  tempera- 
ture) simply  equal  to  the  pressure,  P,  of  the  gas.  This  is  seen 
from  Boyle's  law.  For  when  the  pressure  is  P  and  the  volume 
V,  let  an  additional  small  pressure  p  be  applied  and  let  the 
volume  be  thereby  reduced  by  the  small  quantity  v,  then  by 
Boyle's  Law 

(P+p)(V-v)=PV 

or  if  we  neglect  the  product  of  the  small  quantities  p  and  v 


Now  the  bulk  modulus  is  the  increase  of  pressure  p  divided  by 
the  proportional  decrease  of  volume  v/  V,  and  from  the  last  equa- 
tion this  is  equal  to  P. 

224.  Buoyancy  of  a  Gas.  —  A  body  such  as  a  balloon,  lighter 
than  the  volume  of  air  which  it  displaces,  will  ascend  in  the  air 
when  released.  The  force  giving  it  an  acceleration  upward 
equals  the  difference  of  its  weight  and  the  weight  of  the  air 
which  it  displaces.  If  it  rises  to  such  a  height  that  its  mean 
density  equals  the  density  of  the  rarefied  atmosphere,  it  will 
not  ascend  unless  lightened  by  casting  some  of  its  load  over- 
board. A  large  man  displaces  about  J  Ib.  of  air.  When  a 
body  is  weighed  in  air  with  weights  that  are  supposed  correct 
if  used  in  a  vacuum,  the  true  weight  of  the  body  will  not  be 
obtained  unless  correction  be  made  for  the  effect  of  the  buoyancy 
of  the  air. 


154        MECHANICS  AND  THE  PROPERTIES  OF  MATTER 


Fia.  126.— 
Open  tube 
manometer. 


225.  Manometers. — A  manometer  is  an  apparatus  for  measur- 
ing the  pressure  of  a  fluid.  In  the  simplest  form 
the  pressure  to  be  measured  is  balanced  against  the 
pressure  of  a  column  of  liquid  in  a  tube.  This  is 
called  the  open  tube  manometer  or  siphon  gauge. 
The  pressure  is  found  from  the  difference  of  level  of 
the  liquid  in  the  two  arms  and  the  density,  p,  of  the 
liquid.  In  absolute  units  of  force  P  =  gph  -f  atmos- 
pheric pressure,  while  in  the  weight  of  unit  mass  as 
unit  of  force  P  =  ph  +  atmospheric  pressure. 

In  another  manometer  the  pressure  to  be  mea- 
sured is  balanced  against  that  of  a  gas  (usually  air) 
in  a  uniform  closed  tube.  By  Boyle's  Law  the 
pressure  in  the  gas  is  inversely  as  the  volume,  that 
is,  inversely  as  the  length  of  the  air  column.  The 
pressure  in  the  gas  plus  that  indicated  by  the 
difference  of  level  of  the  liquid  is  the  pressure  to  be 
measured. 

In  Bourdon's  Pressure  Gauge  a  hollow  tube  of 
metal  having  an  elliptical  cross-section  is  bent 
into  an  arc  of  over  180°.  One  end  of  the  tube 
is  closed.  When  the  fluid  of  which  the  pressure 
is  to  be  measured  is  admitted  to  the  open  end, 
the  curved  tube  will  become  less  curved  under 
increased  pressure  and  more  curved  under  de- 
creased pressure.  An  index  moving  over  a  scale 
is  attached  to  the  free  end.  The  action  depends 

on  the  fact  that  the  pressure  tends  to  increase 
the  interior  volume  of  the  tube;  and,  since  a 
circular  cross-section  allows  of  more  volume 
than  an  elliptical  one  for  a  given  periphery, 
the  section  will  under  increased  pressure  tend 
to  the  circular  form  and  the  change  of  form 
of  the  cross-section  causes  the  change  of  shape 
of  the  tube. 

226.  Viscosity  of  Gases. — The  viscosities  of 
gases  are  small  compared  with  those  of  liquids. 
Thus  the  viscosity  of  air  is  about  ^  of  that  of 
water.     While  the  viscosity  of  air  is  small,  it  is  sufficient  to 


Fio.     127 -—Closed 
tube  manometer. 


Fia.   128. — Bourdon's 
pressure  gauge. 


PROPERTIES  OF  GASES  155 

retard  greatly  the  fall  of  small  particles  of  dust  and  small 
drops  of  water  such  as  constitute  a  cloud.  In  a  cloud  (where 
the  air  may  be  one  thousand  times  less  dense  than  water) 
a  drop  of  water  one  thousandth  of  an  inch  in  diameter  falls 
about  0.8  inch  per  second,  while  a  drop  one  ten-thousandth  of 
an  inch  in  diameter  falls  about  one  hundred  times  more  slowly, 
or  about  0.5  inch  in  a  minute.  For  large  drops  such  as  consti- 
tute rain  the  viscosity  of  air  offers  practically  no  resistance;  the 
resistance  which  prevents  such  drops  attaining  enormous  veloci- 
ties is  the  inertia  of  the  air. 

The  viscosity  of  a  gas  increases  when  its  temperature  rises, 
which  is  the  opposite  of  the  case  with  liquids.  The  viscosity  of 
a  gas  at  constant  temperature  does  not  change  appreciably 
when  its  density  is  altered  by  change  of  pressure. 

227.  The  Kinetic  Theory  of  Gases.— The  view  that  a  gas  con- 
sists of  a  myriad  of  particles  in  incessant  motion  may  be  regarded 
as  firmly  established.  The  evidence  for  this  belief  is  that  we  can 
from  it  deduce  nearly  all  the  properties  of  a  gas,  and  the  agree- 
ment between  these  deductions  and  the  observed  facts  could 
hardly  be  a  mere  accidental  coincidence.  As  we  do  not  yet 
know  the  details  of  the  structure  of  the  particles  of  which  a  gas 
consists,  there  are  some  properties  of  a  gas  which  we  cannot  yet 
deduce  from  this  theory.  A  definite  contradiction  between  the 
numerous  known  properties  of  a  gas  and  the  deductions  made 
from  the  theory  would  be  fatal  to  the  latter;  no  such  contradic- 
tion has  ever  been  found. 

As  an  illustration  of  the  way  in  which  the  theory  accounts  for  the  prop- 
erties of  gases  we  shall  show  that  it  explains  Boyle's  Law. 

Before  doing  so  we  must  state  the  theory  more  in  detail.  The  following, 
while  an  incomplete  statement,  will  be  sufficient  for  our  purposes,  (a) 
A  single  gas  consists  of  particles  all  of  the  same  size  moving  in  random 
directions;  (6)  when  the  particles  impinge  on  one  another  and  on  the  walls 
of  the  vessel,  they  rebound  like  smooth  spheres  with  a  coefficient  of  restitu- 
tion of  unity;  (c)  unless  a  gas  is  greatly  condensed,  the  particles  are  so  far 
apart  compared  with  their  dimensions  that  the  forces  they  exert  on  one 
another  may  be  neglected  except  at  impact.  It  will  be  noticed  that  we 
do  not  assume  that  the  velocities  of  all  the  particles  are  the  same, 
and,  in  fact,  there  is  good  ground  for  believing  that  the  velocities  differ 
considerably. 

For  simplicity,  consider  a  gas  contained  in  a  rectangular  vessel  the  edges 
of  which  are  a,  6  and  c  in  length,  and  let  A,  and  A,,  each  of  area  be,  be 


156        MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

perpendicular  to  the  edges  of  length  a.  Let  us  first  fix  our  attention  on 
some  particular  particle  which  has  a  velocity  V  in  some  direction.  V 
may  be  resolved  into  three  components  u,  v,  w,  in  the  directions  of  the 
edges  respectively,  u  being  in  the  direction  of  the  a  sides.  Suppose  the 
particle  to  impinge  on  the  side  Ar  The  force  that  it  will  exert  on  that  side 
at  impact  will  depend  on  its  mass  and  on  u,  not  at 
all  on  v  and  w.  If  it  impinged  without  rebounding 
it  would  give  momentum  equal  to  its  mass,  m,  multi- 
plied by  ut  or  mu.  But  it  rebounds  with  a  velocity 
the  component  of  which  perpendicular  to  A  l  is  u  in 
the  opposite  direction;  hence  the  momentum  it  gives 
to  Al  is  2mu.  Let  us  now  suppose,  for  the  present, 
that  it  reaches  A9  without  impinging  on  any  other 
particle;  for  this  it  will  require  a/u  seconds.  At  A9 

it  will  rebound  with  a  velocity  the  component  of  which  perpendicular  to 
A  2  is  u,  and  will,  supposing  it  to  encounter  no  other  particle,  reach  Al  in 
time  2a/u  when  it  will  again  rebound.  Hence  in  every  second  it  will  im- 
pinge u/2a  times  on  Alf  and  in  every  second  it  will  give  to  At  momentum 
2mu-u/2a,  or  mu9/a.  The  total  force  exerted  on  Alt  that  is  the  momentum 
imparted  to  Al  per  second,  is  the  sum  of  mu*/a  for  all  the  particles,  and, 
to  find  the  pressure  p  on  A,  we  must  divide  this  sum  by  the  area  of  Alt 
namely  be.  Hence 


Let  us  now  denote  the  total  number  of  particles  in  the  vessel  by  N,  and 
the  number  per  unit  volume  by  n.  Since  abc  is  the  total  volume  of  the 
vessel,  nabc  —  N.  Hence 


N 

The  product  mn  is  the  mass  of  all  the  particles  in  unit  volume,  that  is,  the 

u  a  -4-  u  a  h 

density  />;  and  -  — *— —  —  is  the  average  value  of  wa  for  all  the  N  par- 
ticles in  the  vessel.  Denoting  this  by  wa  we  see  that  p=pu*.  For  any 
one  particle 


and,  since  the  particles  are  moving  wholly  at  random,  the  average  values 
of  ua,  v*  and  w*  are  all  equal  and  the  value  of  each  is  therefore  £  of  the 
average  value  of  Fa  which  we  may  denote  by  Fa.  Hence 


If  v  be  the  volume  of  a  mass  M  of  gas,  since  p*-M/v, 


PROPERTIES  OF  GASES  157 

The  total  kinetic  energy  of  translation  of  the  gas  is  the  sum  of  the  kinetic 
energies  of  translation  of  all  the  particles  and  is  evidently  equal  to  \M*V* 
or  f pv.  Now  there  is  good  reason  to  believe  that  if  the  temperature 
of  a  gas  is  constant,  this  kinetic  energy  is  constant.  Hence  the  product 
of  the  pressure  and  volume  of  a  gas  at  constant  temperature  is  constant, 
and  this  is  Boyle's  Law. 

In  the  above  we  have  neglected  the  fact  that  a  particle  may,  during  its 
passages  between  Al  and  A2,  impinge  on  other  particles.  If  such  an  im- 
pact take  place  between  two  particles  moving  along  a  line  perpendicular  to 
Aj  and  A  3,  the  particles  will  exactly  exchange  their  velocities  (§174),  since 
they  are  of  the  same  mass;  and  the  second  particle  will,  therefore,  have  in 
the  a-direction  a  component  equal  to  that  of  the  first  particle  before  impact. 
Thus  the  second  particle  will  take  the  place  of  the  first  in  the  process  described 
above.  When  the  immense  number  of  particles  and  the  random  nature 
of  their  motions  are  considered,  it  is  seen  that  the  effect  is  the  same  as  if 
all  the  impacts  were  in  the  directions  of  u,  v,  and  to. 

The  deviations  from  Boyle's  Law  are  due  to  the  (very  small)  forces 
between  particles  when  they  are  not  in  contact.  These  we  have 
neglected;  by  considering  them  Van  der  Waals  arrived  at  his  more  correct 
law. 


228.  Surface  Condensation  and  Occlusion. — When  a  gas  is  in 
contact  with  a  solid  there  are  molecular  forces  drawing  the  par- 
ticles together,  and  these  produce  more  or  less  condensation  of 
the  gas  on  the  surface  of  the  solid.  This  makes  it  impossible  to 
remove  the  last  traces  of  a  gas  from  a  glass  vessel  by  means  of  an 
air  pump.  It  also  accounts  for  the  fact  that,  when  a  figure  is 
traced  on  a  sheet  of  glass  by  a  stick,  the  figure  will  appear  when 
the  glass  is  breathed  on.  The  breath  condenses  less  readily  on 
the  part  of  the  glass  that  has  been  freed  from  condensed  gas  by 
the  scraping  of  the  stick. 

A  porous  solid  is  readily  permeated  by  a  gas  and  condensation 
on  the  surfaces  of  the  pores  takes  place.  This  is  called  occlusion. 
Very  porous  wood-charcoal  will  absorb  nine  volumes  of  oxygen, 
thirty-five  volumes  of  carbonic  acid  and  ninety  volumes  of  am- 
monia per  volume  of  the  charcoal,  and  cocoanut-charcoal  will 
absorb  still  more.  This  is  why  charcoal  is  so  useful  as  a  deodor- 
izer. Platinum  in  the  porous  form  called  platinum  sponge  will 
absorb  250  times  its  own  volume  of  oxygen.  Palladium  will 
absorb  more  than  one  thousand  volumes  of  hydrogen.  Its  own 
volume  is  thereby  increased  by  about  one-tenth.  The  hydrogen 
is  therefore  reduced  to  one  thousandth  of  its  original  volume;  to 


158       MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

produce  such  a  condensation  by  pressure  alone  would  require  a 
pressure  of  several  tons  per  square  inch. 

229.  Diffusion   of   Gases.  —  Gases,    because    of   their   greater 
mobility,  diffuse  much  more  rapidly  than  liquids.     When  two 
vessels  containing  different  gases  are  connected  by  a  wide  tube, 
diffusion  proceeds  with  great  rapidity,  and  in  a  short  time  each 
gas  is  found  distributed  in  both  vessels  as  if  the  other  gas  were 
not  present.     If  one  of  the  gases  be  a  colored  gas,  such  as  chlorine, 
the  process  of  diffusion  can  be  observed.     As  regards  the  final 
result  each  gas  acts  to  the  other  as  a  vacuum,  but  in  the  process 
of  diffusion  each  gas  retards  the  other.     Gravity  also  plays  some 
part  in  the  process  though  not  in  the  final  result.     Thus  if  the 
gases  be  hydrogen  and  carbon  dioxide,  the  final  mixture  is  at- 
tained more  rapidly  when  the  carbonic  acid  is  in  the  higher 
vessel. 

In  the  process  of  diffusion  of  two  gases  into  each  other 
each  gas  follows  the  same  law  as  holds  for  the  diffusion  of 
two  liquids,  that  is,  each  gas  diffuses  from  places  where  the 
concentration  of  that  gas  is  great  to  places  where  it  is  less, 
and  the  rate  of  diffusion  is  proportional  to  the  rate  of  fall  of 
concentration. 

230.  Efflux  of  Gases.  —  The  rate  of  escape  of  a  gas  through  a 
small  aperture  in  a  very  thin  plate  may  be  deduced  from  the 
principle  of  energy.     Each  part  of  the  gas  as  it  escapes  has  a 
certain  velocity  and  therefore  a  certain  kinetic  energy,  and  this 
must  equal  the  work  performed  by  the  pressure  in  the  vessel  in 
forcing  the  gas  out.     Let  P  be  the  excess  of  the  pressure  in  the 
vessel  over  the  external  pressure.     During  the  escape  of  a  small 
volume  V  of  the  gas  the  pressure  P  does  the  same  amount  of  work 
as  if  it  had  pushed  out  a  piston  in  a  cylinder.     Hence  (§195)  the 
work  done  is  PV.     If  the  density  is  p  the  mass  of  the  volume  V 
of  the  gas  is  Vp,  and  if  its  velocity  is  v  its  kinetic  energy  is  £  Vpv*. 
Equating  the  work  done  to  the  kinetic  energy  which  it  produces, 
we  get 


Thus  the  rate  of  escape  is  directly  as  the  square  root  of  the  pres- 
sure and  inversely  as  the  square  root  of  the  density. 


PROPERTIES  OF  GASES 


159 


Bunsen's  method  of  comparing  the  densities  of  gases  consists 
in  comparing  their  rates  of  escape  through  the  same  aperture 
under  the  same  pressure. 

In  establishing  the  above  formula  we  have  supposed  that  no  work  is 
done  against  internal  friction  such  as  there  would  be  if  the  escape  were 
through  a  tube.  The  wall  of  the  vessel  was  supposed  very  thin  so  that  the 
diameter  of  the  opening  might  be  larger  than  the  thickness  of  the  wall. 
Yet  even  in  this  case  there  is  some  slight  viscous  friction.  This  friction  is 
different  for  different  gases;  hence  the  above  simple  formula  does  not  give 
the  ratio  of  the  densities  very  accurately.  When  a  mixed  gas  escapes  by 
effusion  the  composition  of  the  escpaing  gas  is  not  altered  as  it  escapes. 

When  a  gas  escapes  through  a  porous  partition  in  which  the  pores  are 
very  small,  such  a  fine  unglazed  pottery-ware,  the  circumstances  are 
different  from  those  of  the  above  cases.  The  pores  are  comparable  in  size 
with  the  molecules  of  the  gas  and,  as  might  be  expected,  the  rates  of  escape 
of  different  gases  are  so  different  that  the  constituents  of  a  mixed  gas  escape 
at  different  rates.  This  affords  a  method  of  partially 
separating  the  constituents  of  a  mixed  gas,  and,  as  the 
process  may  be  repeated  several  times,  the  separation  may 
be  made  nearly  complete.  By  this  process  it  has  also  been 
possible  to  show  that  the  molecules  of  a  single  gas  are  all 
of  the  same  sice,  since  no  separation  can  be  produced  by  the 
above  method. 

231.  Passage  of  a  Gas  through  Rubber.  —  Some  gases  also 
escape  through  membranes  such  as  rubber  and  wet  parch- 
ment, in  which  there  are  no  pores  in  the  ordinary  sense. 
The  gas  is  dissolved  by  the  membrane  on  one  side  and 
given  up  on  the  other  side,  so  that  the  passage  through  the 
membrane  is  a  diffusion  from  parts  of  the  membrane  where 
the  concentration  is  greater  to  parts  where  it  Is  less.  The 
same  is  true  of  the  passage  of  a  gas  through  a  film  of  liquid. 
In  a  somewhat  similar  way  hydrogen  will  pass  through 
red-hot  platinum  and  iron. 


mm 


232.  Pumps  for  Liquids.  —  The  oldest  form  of 
pump,  or  suction  pump,  consists  of  a  piston  mov- 
ing  in  a  cylinder  or  barrel  which  is  connected  with 
the  well  by  a  pipe.  In  the  pipe,  or  at  the  top  of  the  pipe,  there 
is  a  valve,  called  the  inlet  valve,  which  can  open  towards  the 
cylinder,  but  not  in  the  opposite  direction;  and  in  the  piston 
there  is  a  valve,  called  the  outlet  valve,  which  can  open  out- 
ward but  not  inward  toward  the  cylinder.  When  the  piston  is 
first  raised  the  air  in  the  cylinder  expands  and  its  pressure 
diminishes.  The  outlet  valve  closes  owing  to  ths  excess  of 


160        MECHANICS  AND  THE  PROPERTIES  OF  MATTER 


pressure  on  the  outside,  and,  for  the  same  reason,  the  inlet 
valve  opens  and  air  from  the  suction  pipe  enters  the  cylinder. 
Thus  the  air  in  the  suction-pipe  is  rarefied  and  the  greater 
atmospheric  pressure  on  the  water  in  the  well  forces  water 
some  distance  up  the  suction-pipe.  After  some  strokes  the 
water  enters  the  cylinder  and  flows  out  by  the  outlet  valve. 

Since  it  is  the  pressure  of  the  atmosphere  that  raises  the  water 
in  a  suction-pump,  water  cannot  be  raised  by  this  means  higher 
in  the  pipe  than  atmospheric  pressure  will  raise  water  in  a  vac- 
uum; and,  since  the  density  of  water  is  to  that  of  mercury  as  1 
to  13.6,  it  follows  that  the  maximum  theoretical  height  is  13.6 
times  the  height  of  the  mercury  in  a  barometer  or  about  34  feet. 
The  practical  limit  of  suction-pumps  is  considerably  less  than 
this,  owing  to  the  presence  of  air  in  water  and  to  the  difficulty 
of  making  the  contact  between  piston  and  pump  air-tight. 
When  water  is  to  be  raised  higher  a  force-pump  is  used  (Fig.  131). 
This  differs  from  the  suction-pump  in  the 
fact  that  the  outlet  valve  is  not  in  the 
piston  but  in  a  side  tube  connected  to 
the  cylinder  near  the  inlet  valve.  During 
each  downward  stroke  of  the  piston  water 
is  forced  up  this  side  tube,  and  the  height 
that  may  be  reached  will  depend  on  the 
force  that  can  be  applied  to  the  piston 
and  the  maximum  pressure  that  the  pump 
will  stand  without  breakage  of  some 
part. 

The  outflow  from  the  delivery  tube  of 
a  force-pump  as  described   above  would 
be  intermittent;  but  it  may  be  rendered 
more  nearly  continuous  by  means  of  an 
"air  chamber,"   in  which   air,  being  put 
""under  pressure  by  the  water  forced   in, 
exerts    continuous    pressure   on   the   out- 
flowing water. 

233.  The  Siphon. — The  siphon  is  a  bent  tube  for  removing 
liquid  from  a  vessel.  The  tube  is  filled  with  liquid  and  is  then 
inverted,  and  one  end  A  is  immersed  in  the  liquid,  while  the  other 
end  C  is  kept  closed.  When  C  is  opened  liquid  flows  through  the 


131. — Force  pump 


PROPERTIES  OF  GASES 


161 


Fio.  132. — Siphon. 


tube  and  out  through  C,  so  long  as  C  is  below  the  level,  D,  of 

the  surface  of  the  liquid. 

To  explain  the  action  of  the  siphon  let  us  consider  the  pressure 

on  the  liquid  at  C  before  the  end  C  is  opened.     If  the  difference 

of  level  of  D  and  C  is  h,  the  pressure  on  the  liquid 

at  C  is  greater  than  atmospheric  pressure  by  gph. 

Hence,  when  C  is  opened,  the  excess  of  pressure 

inside  causes  a  flow,  and  the  flow  continues  so 

long  as  C  is  below  the  level  of  D  and  A  remains 

immersed.     A  siphon  will  not  act  if  the  highest 

point  B  of  the  tube  is  at  a  greater  height  above 

the  level  of  D  than  the  height  to  which  atmos- 
pheric pressure  will  force  the  liquid  in  an  exhausted  tube. 
234.  Air-pumps.  The  first  pump  for  removing  air  from  a  vessel 

was  invented  by  Otto  von  Guericke  (in  1650).  It  was  essen- 
tially a  suction  pump  like  that  used  for  water, 
the  only  difference  being  the  closer  fit  of 
piston  required  to  prevent  leakage  in  the 
case  of  a  gas.  The  degree  of  exhaustion  that 
can  be  attained  by  such  a  pump  is  low.  The 
flap-valve,  at  the  end  of  the  suction-tube,  will 
not  act  automatically  when  the  pressure  in 
the  receiver  has  become  very  small.  For  this 
reason  a  conical  plug,  carried  by  a  rod  that 
passed  with  some  friction  through  the  piston, 
was  substituted.  Another  difficulty  is  caused 
by  the  fact  that  the  piston  cannot  be  made 
to  fit  the  lower  end  of  the  cylinder  with  per- 
fect accuracy,  so  as  to  expel  all  the  air  drawn 
from  the  receiver  into  the  cylinder.  The  latter 
defect  has  been  remedied  in  the  Geryk  pump 
(Fig.  133)  which  has  a  layer  of  oil  at  the  bottom 
of  the  cylinder;  oil  above  the  piston  also  pre- 
vents leakage  at  the  piston  valve. 

235.  High  Vacuum  Pumps. — Many  pumps 
have  been  devised  for  removing  gases  from 
vessels  and  obtaining  very  high  vacua.     In  nearly  all  of  them 
mercury  has  been  used.     In  the  older  forms  the  level  of  mercury 
in  a  large  bulb  connected  to  the  receiver  was  alternately  lowered 
11 


Fio.   133. — Geryk 
pump. 


162        MECHANICS  AND  THE  PROPERTIES  OF  MATTER 


and  raised  so  that  the  gas  was  drawn  from  the  receiver  into  the 
bulb  and  then  ejected  through  a  side  tube,  or  the  mercury  fell 
in  drops  through  a  narrow  tube  and  exerted  suction  on  a  side 
tube  connected  to  the  receiver.  These  forms  of  mercury  pump 
are  rapidly  going  out  of  use  and  we  shall  describe  only  two 
of  the  most  recent  and  efficient  mercury  pumps. 

Langmuir's  "mercury  vapor  pump"1  makes  use  of  the  principle 
of  the  aspirator  (§  193)  in  a  novel  form.  A  current  or  "blast" 
of  mercury  vapor  passes  upward  from  the 
heated  flask,  A,  through  the  tubes  B  and  C 
(Fig.  134)  into  the  condenser  D.  An  annular 
space,  E,  surrounding  B  is  connected  through 
F  with  the  vessel  to  be  exhausted.  C  is  en- 
larged into  a  bulb,  H,  just  above  the  upper 
end  of  B  and  H  is  surrounded  by  a  condenser, 
J,  through  which  water  flows.  The  mer- 
cury condensing  in  D  and  H  returns  to  A 
through  the  tubes  L  and  M.  The  gas  from 
F  passes  freely  up  through  E  and,  meeting 
the  blast  of  mercury  vapor  at  P,  is  blown 
outward  and  upward  along  the  walls  of  the 
condenser  H  and  forced  into  the  main  stream 
of  mercury  vapor  passing  up  through  C  into 
the  condenser  D.  A  less  efficient  pump  con- 
nected to  N  maintains  a  vacuum  of  about 
0.3  mm.  (400  bars)  and  removes  the  gas. 
Langmuir's  pump  will  exhaust  a  vessel  of  11 
liters  capacity  from  atmospheric  pressure  to  a  vacuum  of  0.00001 
mm.  (0.015  bars)  in  80  seconds.  Because  of  its  remarkable  sim- 
plicity and  rapidity  of  action  it  marks  a  great  advance  in  methods 
of  obtaining  high  vacua. 

The  principle  of  Gaede's  mercury  pump  is  indicated  (without  details) 
in  Fig.  135  and  Fig.  136.  An  iron  cylinder,  g,  with  a  glass  face,  g',  is 
more  than  half  filled  with  mercury,  the  surface  of  which  is  at  q.  Inside  of  g 
there  is  a  porcelain  drum,  t,  rotating  about  an  axis,  o,  which  passes  air-tight 
through  g.  This  drum  is  divided  into  two  chambers,  wl  and  wiy  which  com- 

JThe  substance  of  this  description  has  been  supplied,  with  great 
courtesy,  by  the  inventor  of  the  pump,  Dr.  Irving  Langmuir.  A  more 
complete  account  will  appear  in  an  early  number  of  the  Physical  review. 


Fia.  134. —  Langmuirs 
Mercury  vapor  pump. 


REFERENCES 


163 


municate  with  g  by  long  channels  between  the  division-walls  of  t.  Each 
chamber  has  an  opening,  /,  by  which  the  part  of  the  chamber  above  the 
mercury  is  connected,  through  the  tube  R,  with  the  receiver  to  be 
exhausted.  As  the  drum  is  rotated  counter-clockwise,  the  chamber  wl  is 
gradually  emptied  of  mercury  and  filled  with  air  drawn  in  through  R. 
As  the  rotation  continues,/!  is  immersed,  and  the  air  in  wl  is  driven  into  g. 
The  action  of  wa  is  similar.  Since  either  fl  or/2  is  always  out  of  the 


FIG.  135. 


Fia.   136. 


mercury,  the  suction  through  R  is  continuous.  The  air  in  g  is  removed  by 
another  pump  (which  may  be  much  less  efficient)  connected  to  r.  Gaede's 
pump  will  produce  a  vacuum  of  about  0.00004  mm. 


References 

CREW'S  Principles  of  Mechanics  contains  a  brief  and  very  clear  account 
of  the  subject  stated  in  elementary  Vector  and  Calculus  language. 

POYNTINQ  &  THOMSON'S  Properties  of  Matter  is  especially  valuable  for 

information  on  gravitation,  elasticity  and  properties  of  fluids. 
The  above-mentioned  books  will  be  found  useful  for  somewhat  advanced 

systematic  study. 

MACH'S  Principles    of   Mechanics  is  a   very  interesting  and  elementary 
account  of  the  historical  development  of  the  subject. 

Cox's  Mechanics  is  an  elementary  book  with  notes  on  the  historical  de- 
velopment. 

WORTHINGTON'S  Dynamics  of  Rotation  is  an  elementary  book  with  numerous 
suggestive  experiments. 

PERRY'S  Spinning  Tops  is  a  popular  account  of  the  principles  of  the  gyro- 
scope. 

LOVE'S  Theoretical  Mechanics  is  a  very  careful  account  of  the  logical  relations 
of  the  parts  of  the  subject. 

MAXWELL'S  Matter  and  Motion,  while  elementary  and   very  brief,  is  a 
masterpiece  by  a  great  physicist. 

TAIT'S  Properties  of  Matter  contains  an  elementary  treatment  of  gravitation, 
elasticity  and  properties  of  fluids. 


164        MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

LenxiBj's  Pioneers  of  Science  consists  of  popular  lectures  on  Galileo,  Newton, 

etc. 

POYNTING'B  Mean  Density  of  the  Earth  describes  all  the  methods  used. 
Encyclopedia  Britannica,  articles  on  "  Weights  and  Measures,"  "  Mechanics," 

"Elasticity,"  etc. 
DANIELL'S  Principles  of  Physics  is  a  large  compendium. 

Problems 

1.  A  train  acquires  5  minutes  after  starting  a  velocity  of  40  km.  per  hour. 

Assuming    constant   acceleration,  what  is  the  distance  passed    over 

V  1     'tv      d        during  the  5th  minute?  Ans.  0.6  km. 

.       .      ..  2.  A  train  having  a  speed  of  70  km.  per  hour  is  brought 

to  rest  by  brakes  in  a  distance  of  600  m.     What  is 

the  acceleration  (assumed  constant)?  Ans.  —1.13  km./min1. 

3.  What  is  the  final  speed  of  a  body  which,  moving  with  uniform  accelera- 
tion travels  72  meters  in  2  minutes  if: 

(a)  the  initial  speed  =  0? 

(b)  the  initial  speed  =  15  cm.  per  sec.? 

Ans.  120  cm. /sec.;  105  cm. /sec. 

4.  A  body  is  projected  at  an  angle  of  30°  with  the  horizon  with  a  velocity 
of  30  m.  per  sec.     When  and  where  will  it  again  meet  this  horizontal 
plane?     How  far  will  it  ascend?  Ans.  3.06  s.;  79.5  m.;  11.4  m. 

5.  A  body  slides  down  an  inclined  plane  and  in  the  3d  sec.  describes 
110  cm.     What  is  the  inclination?  Ans.  2°  35'. 

6.  What  initial  vertical  velocity  must  a  ball  have  in  order  to  fall  back 
to  its  starting  point  in  10  sec.?  Ans.  4900  cm  /sec. 

7.  At  what  angle  with  the  shore  must  a  boat  be  directed  in  order  to 
reach  a  point  on  the  other  shore  directly  opposite,  if  the  speed  of  the 
boat  be  4  miles  per  hour  and  that  of  the  stream  be  2  miles  per  hour? 

Ans.  60°. 

8.  A  point  goes  over  a  circular  path  10  cm.  in  diameter  4  times  a  second, 
at  a  uniform  speed.     To  what  acceleration  is  it  subject? 

Ans.  3158  cm. /sec*. 

9.  A  ball  rises  to  a  height  of  50  ft.  and  travels  200  ft.  horizontally.     Find 
the  direction  and  magnitude  of  the  velocity  with  which  it  is  thrown 

Ans.  0  =  45°;  v  =  80.2  ft. /sec 

10.  Show  that  the  time  of  descent  (without  friction)  down  all  chords  of 
a  vertical  circle  is  the  same. 

11.  What  velocity  must  a  boy  give  a  sling  of  80  cm.  radius  in  order  that 
the  stone  shall  not  fall  out  of  the  sling?  Ans.  280  cm. /sec. 

12.  What  force  will  a  man  who  weighs  70  kg.  exert  upon  the  floor  of  an 
elevator  descending  with  an  acceleration  of  100  cm.  per  sec.  per  sec.? 

_  If  ascending  with  the  same  acceleration? 

^  Ans.  62.8  kg.  wt.;  77.1  kg.wt. 

13.  A  force  of  1000  dynes  acts  upon  a  mass  of  1  kg.  far 

1  min.     Find  the  velocity  acquired  and  the  space  passed  over  in  this 

time.  Ans.  60  cm./sec.;  1800  cm. 


PROBLEMS  165 

14.  A  shot  weighing  10  Ibs.  is  shot  from  a  gun  weighing  3  tons  with  an 
initial  velocity  of  1200  feet  per  seo.     What  is  the  initial  velocity  of 
the  recoil?  Ana.  2  ft./sec. 

15.  Three  forces,  5,  12,  15  are  in  equilibrium.     Find  the  angles  between 
them.  Ans.  62°  11';  134°  58';  162°  51'. 

16.  Bodies  of  mass  10  kg.  and  8  kg.  are  connected  by  a  string  over  a  pulley. 
How  far  does  each  move  from  rest  in  the  first  two  seconds? 

Ans.  218  cm. 

17.  Twelve  bullets  are  divided  between  two  scale  pans  connected  by  a 
cord   passing   over  a   pulley.     What   distribution   will   produce   the 
greatest  tension  on  the  support  of  the  pulley? 

18.  A  cord  passes  over  two  fixed  pulleys  and  through  a  third  pulley  sus- 
pended between  them.     A  mass  of  10  g.  is  attached  to  one  end  of  the 
cord,  a  mass  of  5  g.  to  the  other  end,  and  the  suspended  pulley  and  the 
attached  weight  weigh  2  g.     The  cords  being  all  vertical,  what  are  the 
accelerations  of  the  three  masses?  Ans.  809  cm./sec2.;  639;  724. 

19.  A  baseball  whose  mass  is  300  g.  when  moving  with  a  velocity  of  20  m. 
per  sec.  is  squarely  struck  by  a  bat  and  then  has  a  velocity  of  30  m.  per 
sec.  in  the  other  direction.     Calculate  the  impulse  and  average  force  if 
the  contact  last  .02  sec.         Ans.  15X10J  g.  — cm./sec.;  7.5X107 dynes. 

20.  With  how  much  energy  must  a  bullet  weighing  20  g.  be  shot  hori- 
zontally from  a  gun  4  m.  above  a  level  plane,  in  order  to  strike  the 
W    k       t\       ground  300  m.  away  from  the  gun? 

Energy?11  An8'  1-11><1010  erSs- 

21.  A  projectile  traveling  at  the  rate  of  700  ft.  per  sec. 

penetrates  to    the  depth  of  2  in.     Find  the  velocity  necessary  to 
penetrate  3  in.  Ans.  857  ft./sec. 

22.  A  hammer  weighing  6  kg,  and  moving  with  a  velocity  of  100  cm.  per 
sec.  drives  a  nail  into  a  plank  1  cm.     What  resistance  does  it  overcome 
(supposed  uniform)?  Ans.  3X107  dynes. 

23.  A  man  can  bicycle  12  miles  an  hour  on  a  smooth  road;  downward 
force  of  each  foot  in  turn  =  20  Ibs.,  length  of  stroke  =  1  ft.,  bicycle  is 
advanced  17  ft.  for  each  revolution  of  the  cranks.     At  what  H.  P.  does 
he  work?  Ans.  .075  H.  P. 
A  man,  weight  180  Ibs.,  runs  up  26  steps,  each  7  in.  high,  in  4  sec. 
At  what  H.  P.  does  he  work?  Ans.  1.2  H.  P. 

25.  A  sprinter  who  weighs  161  Ibs.  runs  40  yds.  in  4f  sec.,  60  yds.  in  6f  eec., 
100  yds.  in  10  sec.     What  is  (a)  his  velocity  from  40  to  60  yds.  and 
from  60  to  100  yds.,  (b)  his  kinetic  energy  at  the  end  of  40  yards? 
(c)  Calculate  the  rate  of  working  in  H.  P.  required  to  produce  this 
kinetic  energy,    (d)  In  what  other  ways  does  he  expend  energy? 

Ans.  (a)  33$  ft./sec.     (b)  2777  ft.  Ibs.      (c)  1.09  H.  P. 

26.  Find  the  number  of  watta  in  one  horse-power.  Ans.  746. 

27.  A  sprinter  does  100  yards  on  the  horizontal  in  10.5  sec.,  and  the  same 
distance  up  hill  with  a  rise  of  32  ft.  in  17.5  sec.     Assuming  that  his  rate 
of  working  is  the  same  throughout,  calculate  the  added  work  done  in 
the  additional  7.0  seconds  up  hill  and  the  rate  of  working  that  this 
hnpli«8,  the  man's  weight  being  160  Ibs.  Ans.  1.33  H.  P. 


166        MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

28.  100  cu.  ft.  of  water  pass  over  a  dam  10  ft.  high  in  1  min.     What  horse- 
power could  be  derived  from  this  if  all  were  utilized?       Ans.  1.9  H.  P. 

29.  A  30-gram  rifle  bullet  is  fired  into  a  suspended  block  of  wood  weighing 
15  kilos.     If  the  block  is  suspended  by  a  string  of  length  2  meters  and 
is  moved  through  an  angle  of  20°,  calculate  the  velocity  of  the  bullet. 
Notice  that  the  impact  of  the  bullet  on  the  block  does  not  change  the 
total  momentum  of  both  (§  46)  and  during  the  subsequent  swing  of  the 
oendulum  its  total  energy  remains  constant.  Ans.  770  m./sec. 

30.  If  a  locomotive  driving-wheel  1.5  m.  in  diameter  makes  250  revolutions 
per  minute,  what  is  the  mean  linear  speed  of  a  point  on  the  periphery? 
P        .  Of  the  point  when  it  is  highest?     When  it  is  lowest? 

Ans.  19.6  m./sec.;  39.2  m./sec.;  0. 

31.  The  armature  of  a  motor  revolving  at  the  rate  of  1800  revolutions 
per  minute  comes  to  rest  in  20  seconds  after  the  current  is  shut  off. 
Calculate  its  average  angular  acceleration  and  the  number  of  revolu- 
tions. Ans. -9  42  rad./sec.2;  300  rev. 

32.  Find  in  radians  per  second  the  angular  velocity  of  the  earth  about  its 
axis  and  deduce  the  component  of  this  angular  velocity  about  a  diam- 
eter through  a  point  in  latitude  40°.     (Principle  of  Foucault's  pendulum) . 

33  A  circle  has  a  diameter  of  16  cm.  A  smaller  circle  tangent  to  it  and 
12  cm.  in  diameter  is  cut  out  of  it.  Where  is  the  center  of  gravity 
c  .  of  the  remainder? 

„  Ans.  10.6  cm.  from  common  tangent. 

34.  Two  cylinders  of  equal  length  (=-20  in.),  and  having 

diameters  of  12  and  6  in.,  are  joined  so  that  their  axes  coincide.     Where 

is  the  center  of  gravity?  Ans.  6  in.  from  junction. 

35.  Find  the  center  of  gravity  of  a  table  4  ft.  X3  ft.  Xl  in.,  with  legs  at 
the  corners  2  ft.  X2  in.  X2  in.  Ans.  0.233  ft.  from  top. 

36.  The  mass  of  the  moon  is  -fa  of  that  of  the  earth  and  the  average  distance 
between  their  centers  is  240,000  miles.     Calculate  the  position  of  the 
center  of  mass  of  the  two.  Ans.  2963  m.  from  center  of  earth. 

37.  At  the  corners  of  an  equilateral  triangle  ABC  masses  of  1,  2  and  3  Ibs. 
respectively  are  placed.    Find  the  distance  of  their  center  of  mass 
from  BC  assuming  each  side  of  the  triangle  to  be  1  ft.  in  length. 

Ans.  0.144  ft. 

38.  A  bar  6  ft.  long  and  pivoted  at  the  middle  has  a  weight  of  24  Ibs. 
hung  at  one  extremity.     What  is  the  moment  of  the  weight  (a)  when 
__  the  bar  is  horizontal,  (b)  when  it  makes  an  angle 

of  30°  below,  and  (c)  of  60°  above  with  the  horizontal 
position?  Ans.  72;  62.3;  36  Ibs.  wt.  ft.. 

39.  If  it  is  wished  to  upset  a  tall  column  by  a  rope  of  given  length  pulled 
from  the  ground,  where  should  it  be  applied,  if  the  length  of  the  rope 
is,  (1)  equal  to,  (2)  twice,  the  height  of  the  column? 

40.  Find  the  moment  of  inertia  of  a  sphere  (m  =  20,  r«2)  about  an  axis 
tangent  to  its  surface.  Ans.  112. 

41.  Find  the  moment  of  inertia  of  three  circular  disks,  all  three  touching 
each  other  in  the  same  plane,  about  a  perpendicular  axis  passing  through 


PROBLEMS  167 

the  center  of  one  of  them.     The  mass  of  each  is  100  g.  and  the  radius 
of  each  is  6  cm.  Ans.  34,200  gin.  cm.a 

42.  Two  masses,   100  kg.  and  200  kg.,  respectively,  are  connected  by  a 
rigid  rod  1  m.  long.     The  system  is  thrown  so  that  the  center  of 
gravity  has  a  velocity  of  20  m.  per  second  and  the  system  turns  10 
times  per  second  about  this  center.     Find  the  kinetic  energy  of  the 
system.  Ans.  192  X 1010  ergs. 

43.  What  energy  has  a  grindstone  1£  m.  in  diameter,  weighing  1000  kg. 
and  rotating  once  every  2  sec.?  Ans.  13.9 X109  ergs. 

44.  A  solid  iron  cylinder,  100  cm.  diameter,  rolls  down  a  plane  6  m.  long 
inclined  at  30°.     What  linear  velocity  does  it  acquire? 

Ans.  627  cm. /sec. 

46.  A  block  of  stone  weighs  2.5  tons  and  is  in  the  form  of  a  cube  of  1  yard 
side.  It  rests  on  level  ground.  What  is  the  least  force  which  applied 
to  the  block  will  cause  it  to  revolve  about  a  horizontal  edge? 

Ans.  1768  Ibs.  wt. 

46.  Parallel  forces  of  1,  2  and  3  units  respectively  act  at  the  corners  A, 

B,  C  of  an  equilateral  triangle  of  1  ft.  side.    Find  the  distance  of  the 

resultant  from  BC.  Ans.  0.144  ft. 

47.  Parallel  forces  of   10  and  6,  but  in  opposite 

directions,  are  applied  to  a  bar  at  distances  of  8  and  3  from  one  end. 

What  is  the  magnitude  of  the  resultant  and  where  does  it  act? 

Ans.  4;  15.5. 

48.  Two  equal  parallel  forces,  each  50  dynes,  act  in  opposite  directions 
at  the  ends  of  a  bar  10  cm.  long.     The  bar  makes  an  angle  of  45°  with  the 
direction  of  the  force.     What  is  the  moment  of  the  couple? 

Ans.  353.5  dynes-cm. 

49.  A  man  and  a  boy  carry  a  weight  of  20  kg.  between  them  by  means  of 
a  uniform  pole  2  m.  long,  weighing  5  kg.     Where  must  the  weight  be 
placed  so  that  the  man  may  carry  twice  as  much  of  the  whole  weight  as 
the  boy?  Ans.  0.416  m.  from  middle. 

50.  A  rod,  the  mass  of  which  is  1  kg.,  hangs  from  a  hinge  on  a  vertical 
wall  and  rests  on  a  smooth  floor.     Calculate  the  force  on  the  floor  and 
_,     ....   .  the  force  on  the  hinge.  Ans.  500  g.;  500  g. 

L'  61.  A  uniform  ladder  30  feet  long  and  of  50  Ibs.  weight 
rests  with  the  upper  end  against  a  smooth  vertical  wall,  and  the  lower 
end  is  prevented  from  slipping  by  a  peg.  If  the  inclination  of  the 
ladder  to  the  horizontal  is  30°,  find  the  force  on  the  wall  and  at  the 
peg.  Ans.  43.3  Ibs.  wt.;  66.1  Ibs.  wt. 

52.  A  barn  door  is  10  ft.  long  and  5  ft.  wide  and  weighs  200  Ibs.     The 
hinges  are  1  ft.  from  the  ends  and  the  weight  is  carried  entirely  by  the 
upper  hinge.     Find  the  direction  and  magnitude  of  the  resultant  force 
on  the  upper  hinge.  Ans.  209  Ibs.  wt.;  17°  21'  to  vertical. 

53.  One  end  of  a  certain  rod  is  clamped.      If  the  other  end  is  pulled  1  cm. 
from  its  natural  position  and  then  released,  it  starts  with  an  acceleration 

-.  of  10  cm.  per  sec.  per  sec.     Wrhat  is  the  period  of  ite 

Periodic  motions.     ,,      .,     ;  A       t  oc 

vibration?  Ans.  1.98  sec. 


168        MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

54.  The  balance-wheel  of  a  watch  makes  5  complete  vibrations  in  2  sec. 
With  what  angular  acceleration  will  it  start  when  turned  30°  from  its 
position  of  equilibrium  and  released?  Ana.  129.34  rad./sec8. 

55.  A  hoop  of  25  cm.  radius  hangs  on  a  peg.    Prove  that  its  period  of 
vibration  is  equal  to  that  of  a  simple  pendulum  whose  length  is  equal 
to  the  diameter  of  the  hoop. 

56.  A  clock  gains  3  min.  a  day.     Find  the  error  in  the  length  of  the  pendu- 
lum, regarding  it  as  a  simple  pendulum.  (<7=»9SO).          Ans.  0.414  cm. 

57.  A  pendulum  which  is  a  seconds  pendulum  where  g  =»  980,  vibrates  but 
59.95  times  a  minute  on  top  of  a  mountain.     What  is  the  acceleration 
of  gravity  at  this  point?  Ans.  978.37. 

58.  A  rod  2  m.  long  is  freely  suspended  at  one  end.     Calculate  its  period 
of  vibration.  Ans.  2.31  sec. 

59.  A  seconds  pendulum  is  drawn  aside  and  released  and  at  the  same 
moment  a  ball  is  allowed  to  fall.     The  ball  and  the  bob  collide  as  the 

j    pendulum  passes  through  the  vertical.     Calculate  the  height  of  fall  of 
the  ball.  Ans.  122.5  cm. 

60.  The  coefficient  of  friction  for  two  surfaces -0.14.     A  pull  of  20  kg. 
weight  will  overcome  what  pressure  between  them?  Ans.  143  kg. 

„  .    .  61.  What  force  applied  parallel  to  a  plane  inclined  at  20° 

will  push  up  a  block  weighing  100  kg.,  the  coefficient  of 
friction  between  the  two  being  0.24;  (a)  the  block  moving  uniformly; 
(6)  the  block  having  an  acceleration  of  100  cm.  per  sec.  per  sec.? 

Ans.  (a)  56.7  kg.  wt.;  (6)  66.9  kg.  wt. 

62.  What  is  the  coefficient  of  friction  between  a  body  and  a  horizontal 
plane  if  the  body  loses  a  velocity  of  100  ft.  per  sec.  and  comes  to  rest  in 
moving  200  ft.  over  the  plane?  Ans.  0.776. 

63.  A  toboggan  slides  100  yards  down  a  track  inclined  at  20°  to  the  horizontal 
in  11  seconds.     Calculate  the  coefficient  of  friction.  Ans.  0.20. 

64.  A  small  block  rests  on  a  horizontal  revolving  platform  at  a  distance 
of  40  cm.  from  the  axis  of  revolution.     If  the  coefficient  of  friction 
is  .30  at  what  angular  velocity  of  the  platform  will  the  block  just  begin 
to  slip?  Ans.  2.71  rad./sec. 

65.  A  man  raises  a  stone  1  in.  with  a  lever  of  the  first  class  10  ft.  long 
weighing  50  Ibs.,  the  fulcrum  being  1  ft.  from  the  point  of  application  to 
._    .  .  the  stone.    If  he  exerts  a  force  of  100  Ibs.  wt.  what 

force  is  applied  to  the  stone  and  what  work  does  he 
do?  Ans.  1,100  Ibs.  wt.;  75  ft.  Ibs. 

66.  A  boy  who  exerts  a  push  of  50  Ibs.  wt.  wishes  to  roll  a  barrel  weighing 
200  Ibs.  into  a  wagon  2£  ft.  high.     Assuming  that  he  pushes  in  a 
line  through  the  center  of  the  barrel  parallel  to  the  plank,  how  long  a 
plank  will  he  need  and  how  much  work  will  he  do? 

Ans.  10  ft.;  500  ft.  Ibs. 

67.  A  body  weighs  12  Ibs.  on  one  side  of  a  false  balance  and  12.5  Ibs.  on 
the  other  side.     What  is  the  ratio  of  the  arms  of  the  balance? 

Ans.  1.021. 
68k  A  man  weighing  150  Ibs.  sits  on  a  platform  suspended  from  a  movable 


PROBLEMS  169 

pulley  and  raises  himself  by  a  rope  passing  over  a  fixed  pulley.     Sup- 
posing the  oords  are  parallel,  what  force  does  he  exert? 

Ana.  50  Ibs.  wt. 

69.  A  wheel  whose  radius  is  25  cm.  is  fastened  to  one  end  of  a  screw  whose 
pitch  is  1  mm.     What  force  can  the  screw  exert  in  its  nut  when  a  force 
of  1  kg.  wt.  is  applied  tangentially  to  the  wheel,  friction  being  supposed 
negligible?  Ans.  1570  kg.  wt. 

70.  Compare  the  mechanical  advantages  of  a  block  and  tackle  when  the 
end  of  the  cord  is  attached  to  the  upper  block  and  when  it  is  attached 
to  the  lower. 

71.  How  far  above  the  surface  of  the  earth  must  a  body  be  to  lose  0.1  per 
cent,  in  weight?  Ans.  1.95  mi. 

,,      .     .  72.  If  the  moon's  mass  is  -fa  that  of  the  earth,  and  its 

diameter  2160  miles,  that  of  the  earth  being  7900 
miles,  what  is  the  acceleration  of  gravity  on  the  moon's  surface? 

Ans.  164  cm./sec*. 

73.  Find  the  time  of  revolution  of  the  earth  which  would  cause  bodies  to 
have  no  apparent  weight  at  the  equator.  Ans.  1.41  hr. 

74.  A  wire  300  cm.  long  and  1  mm.  in  diameter  is  stretched  1  mm.  by  a 
weight  of  3000  g.     What  is  Young's  Modulus? 

Ans.  11.2  XlOn  dynes/cm.1. 

76.  A  weight  is  hung  from  the  ceiling  by  a  steel  wire  2  m. 

Elasticity.  long  and  of  1  mm.  diameter  joined  to  a  copper  wire 

1  m.  long  and  of  0.5  mm.  diameter.     Another  weight 

sufficient  to  produce  a  total  extension  of  1  mm.  is  added.     Calculate  the 

extension  of  each  part.  Ans.  0.19  mm.;  0.81  mm. 

76.  To  opposite  faces  of  a  cubical  block  of  jelly  of  20  cm.  edge  parallel  and 
opposite  forces  of  1  kg.  each  are  applied  and  produce  a  relative  motion 
of  1  cm.     Calculate  the  strain,  the  stress  and  the  shear  modulus. 

Ans.  0.05;  2450  dynes/cm.a;  49000  dynes/cm.2 

77.  An  iron  bar  of  400  c.c.  volume  falls  from  a  ship  and  sinks  to  the  bottom 
of  an  ocean  1000  m.  deep.     How  much  is  its  volume  diminished, 
assuming  that  each  10  m.  of  water  pressure  produces  a  pressure  equal  to 
that  of  the  atmosphere,  which  equals  one  million  dynes  per  sq.  cm. 

Ans.  0.026  c.c. 

78.  A  ball  weighing  20  kg.,  moving  with  a  velocity  of  500  cm.  per  sec., 
strikes  a  second  ball  weighing  100  kg.  which  is  at  rest.     If  the  first  ball 
rebounds  with  a  velocity  of  100  cm.  per  sec.,  what  will  be  the  velocity  of 
the  second?  Ans.  120  cm./sec. 

79.  Two  bodies  differing  in  bulk  weigh  the  same  in  water;  compare  the 
weights  in  mercury;  in  vacuo. 

P         ti       *    80.  A  mass  of  copper  suspected  of  being  hollow  weighs 

523  g.  in  air  and  447.5  g.  in  water.     What  is  the  volume 

of  the  cavity?  Ana.  16.8  c.c. 

81.  The  specific  gravity  of  ice  is  0.918,  that  of  sea-water  1.03.     What  is  the 

total  volume  of  an  iceberg  of  which  700  cu.  yds.  is  exposed? 

Ans.  6438  cu.  yds. 


170        MECHANICS  AND  THE  PROPERTIES  OF  MATTER 

82.  A  block  of  wood  weighing  1  kg.,  whose  specific  gravity  is  0.7,  is  to  be 
loaded  with  lead  so  as  to  float  with  0.9  of  its  volume  immersed.     What 
weight  of  lead  is  required  (1)  if  the  lead  is  on  top?     (2)  if  the  lead  is 
below?  Ans.  286  g.;  313.5  g. 

83.  A  hydrometer  sinks  to  a  certain  mark  in  a  liquid  of  sp.  gr.  0.6,  but  it 
takes  120  g.  to  sink  it  to  the  same  mark  in  water.     What  is  the  weight 
of  the  hydrometer?  Ans.  180  g. 

84.  One  of  the  limbs  of  a  U-shaped  glass  tube  contains  mercury  to  the 
height  of  0.175  m.;  the  other  contains  a  different  liquid  to  a  height  of 
0.42  m.,  the  two  columns  being  in  equilibrium.     Required,  the  specific 
gravity  of  the  second  liquid  with  reference  to  mercury  and  to  water. 

85.  Find  the  volume  in  cu.  ft.  of  the  smallest  block  of  ice  which,  floating 
on  fresh  water,  will  just  carry  a  man  who  weighs  150  Ibs. 

Ans.  29.3  cu.  ft. 

86.  Given  a  body  A  which  weighs  7.55  g.  in  air,  5.17  g.  in  water,  and  6.35  g. 
in  another  liquid  B;  required,  the  specific  gravity  of  the  body  A  and  the 
liquid  B.  Ans.  3.17;  0.504. 

87.  A  block  of  brass  10  cm.  thick  floats  on  mercury.     How  much  of  its 
volume  is  above  the  surface,  and  how  many  cm.  of  water  must  be  poured 
above  the  mercury  so  as  to  reach  the  top  of  the  block?     (Density  of 
mercury  — 13.6;  of  brass  =  8.5.)  Ans.  0.375  of  the  whole;  4.05  cm. 

88.  Two  tubes  are  inserted  in  a  vessel  of  water  on  the  same  horizontal  plane. 
The  diameter  of  the  one  is  0.5  mm.  and  its  length  is  20  cm.;  the  diameter 
of  the  other  is  0.25  mm.  and  its  length  is  10  cm.     Compare  the  amounts 
of  water  flowing  through  the  two  tubes  in  a  given  time.        Ans.  8:1. 

89.  The  diameter  of  the  small  piston  of  an  hydrostatic  press  is  2  in.,  the 
diameter  of  the  large  piston  is  2  ft.     What  weight  on  the  small  piston 
will  support  two  tons  on  the  large  piston?  Ans.  27.77  Ibs. 

90.  The  pressure  at  the  bottom  of  a  lake  is  three  times  that  at  a  depth  of 
2  m.     What  is  the  depth  of  the  lake?     (Atmospheric  pressure  =  76  cm. 
of  mercury.)  Ans.  26.67  m. 

91.  A  retaining  wall  3  m.  wide  and  40  m.  long  is  inclined  at  30°  to  the 
horizontal.    Find  the  total  force  in  kg.  exerted  against  it  by  the  water 
when  the  water  rises  to  the  top.  Ans.  9Xl04kg. 

92.  What  is  the  outward  force  exerted  by  the  water  on  the  sides  of  a  cir- 
cular tank  1  m.  in  "diameter,  the  height  of  the  water  being  150  cm.? 
What  is  the  thrust  due  to  the  water  on  the  bottom? 

Ans.  3532  kg.  wt.;  1178  kg.  wt. 

93.  The  surface  tension  of  a  soap-bubble  solution  is  27.45  (dynes/cm.). 
How  much  greater  is  the  pressure  inside  a  soap-bubble  of  3  cm.  radius 
than  in  the  air  outside?  Ans.  36.6  dynes/cm3. 

94.  How  far  will  water  be  projected  horizontally  from  an  aperture  3  m. 
below  the  water  level  of  a  tank  and  10  m.  above  the  ground  (neglecting 
air  resistance)?  Ans.  10.96  m. 

95.  A  body  whose  specific  gravity  is  2  is  weighed  in  air  of  specific  gravity  0.0013 
with  weights  of  specific  gravity  9.     The  weight  in  air  beiixg  100  g.,  what  js 
the  true  weight?  ATM.  100.050    . 


PROBLEMS  171 

p          .  96.  If  the  barometer  sinks  15  mm.,  how  much  is  the 

pressure  in  dynes  per  sq.  cm.  decreased? 

Ans.  19992  dynes/cm'. 

97.  An  air  bubble  at  the  bottom  of  a  pond  6  m.  deep  has  a  volume  of  0.01 
c.c.     Find  the  volume  just  as  it  reaches  the  surface,  the  barometer 
standing  760  mm.  Ans.  0.0158  c.c. 

98.  Owing  to  the  presence  of  air  the  mercury  column  in  a  barometer  85  cm. 
long  stands  at  70  cm.  when  an  accurate  barometer  stands  at  75  cm. 
What  pressure  will  this  barometer  indicate  when  an  accurate  barometer 
stands  at  72  cm.?  Ans.  67.67  cm. 

99.  A  barometer  reads  73  cm.     Calculate  the  thrust  on  one  side  of  a  board 
1  m.  square.  Ans.  9928  kg.  wt. 

100.  A  barometer  has  a  cross-section  of  2  sq.  cm.  and  is  so  long  that  as  the 
mercury  stands  at  76  cm.,  there  is  a  vacuum  space  10  cm.  long. 
Some  air  is  allowed  to  enter  and  the  mercury  falls  10  cm.     What  was 
the  volume  of  the  air  before  it  entered?  Ans.  5.26  cm». 

101.  How  high  must  we  ascend  above  the  sea-level  to  observe  a  depression 
of  1  mm.  in  the  height  of  the  barometer?     Density  of  air  — 0.0013 
(approx.).  Ans.  10.4  m. 

102.  A  glass  tube  60  cm.  long,  closed  at  one  end,  is  sunk,  open  end  down,  to 
the  bottom  of  the  ocean.     Wrhen  drawn  up  it  is  found  that  the  water 
has  penetrated  to  within  5  cm.  of  the  top.     Atmospheric  pressure 
—  76  cm.  of  mercury.     Calculate  the  depth  of  the  ocean,  assuming  the 
density  constant,  and  equal  to  1.026.     (Principle  of  Lord  Kelvin's 
sounding  apparatus.)  Ans.  110.8  m. 

103.  In  a  vessel  of  1  cu.  meter  volume  are  placed  the  following  amounts  of 
gas:  (1)  hydrogen,  which  occupies  1  cu.  m.  at  atmospheric  pressure. 

(2)  nitrogen,  which  occupies  3  cu.  m.  at  a  pressure  of  2  atmospheres. 

(3)  oxygen,  which  occupies  2  cu.  m.  at  a  pressure  of  3  atmospheres. 
Calculate  pressure  of  mixture.  Ans.  13  at. 

104.  The  mouth  of  a  vertical  cylinder  18  in.  high  is  closed  by  a  piston  whose 
area  is  6  sq.  in.     If  a  weight  of  100  Ibs.  be  placed  on  the  piston,  how 
far  will  it  descend,  supposing  the  atmospheric  pressure  to  be  14  Ibs. 
per  sq.  in.,  the  friction  negligible  and  the  temperature  constant? 

Ans.  9.8  in. 

105.  A  cylindrical  diving-bell  7  ft.  in  height  is  lowered  until  the  top  of  the 
bell  is  20  ft.  below  the  surface  of  the  fresh  water.     If  the  barometer 
height  at  the  time  is  30  in.,  how  high  will  the  water  rise  in  the  bell? 
What  air  pressure  in  the  bell  would  just  keep  the  water  out? 

Ans.  2.96  ft.;  1.82  at. 

106.  (a)  What  fraction  of  an  atmosphere  is  the  difference  in  pressure  be- 
tween two  points  in  air  at  0°  C.  and  76  cm.  pressure  if  the  difference 
of  level  is  1  cm.?    (b)  How  large  a  difference  of  level  would  produce 
a  difference  of  pressure  of  0.01  per  cent,  of  an  atmosphere? 

Ans.  126X10-*;  80  cm. 


WAVE  MOTION 

BY  E.  PERCIVAL  LEWIS,  PH.  D. 
Professor  of  Physics  in  the  University  of  California 

236.  Characteristics  of  Wave  Motion. — The  word  wave  recalls 
the  familiar  phenomena  observed  whenever  the  surface  of  a  body 
of  water  is  disturbed.    Large  waves  are  usually  so  irregular  that  it 
would  be  difficult  to  reach  any  general  conclusions  regarding  the 
laws  of  their  formation  or  propagation.     If  less  complex  waves 
be  observed,  such  as  those  produced  by  throwing  a  pebble  into  a 
quiet  pond  or  by  the  gentle  disturbance  of  the  water  or  mercury 
in  a  tank,  it  will  be  seen  that  they  are  alternate  ridges  and  hollows 
in  the  surface,  which  diverge  in  uniformly  expanding  circles  from 
the  center  of  disturbance.     If  small  pieces  of  cork  rest  on  the  sur- 
face another  important  characteristic  of  wave  motion  may  be 
observed.     The  particles  rise  on  an  approaching  wave,  ride  for- 
ward on  its  crest  for  a  short  distance,  then  fall  back  into  the  suc- 
ceeding hollow,  to  again  move  upward  and  forward  on  the  next 
crest.     They  describe  orbits  in  a  vertical  plane  which  are  evidently 
circular  or  elliptical.     Since  these  particles  participate  in  the 
movement  of  the  water  on  which  they  rest,  it  is  plain  that  the 
water  as  a  whole  does  not  move  continuously  forward  with  the 
waves,  but  that  each  element  rotates  about  its  original  undis- 
turbed position,  to  which  it  returns  when  the  train  of  waves  has 
passed.     Waves  are,  therefore,  the  progression  of  a  shape  or 
condition,  not  of  matter. 

237.  Water  waves  illustrate  the  following  fundamental  charac- 
teristics of  all  wave  motions  in  material  media:  (1)  All  parts  of 
the  medium  reached  by  the  disturbance  are  subject  to  periodic  dis- 
placements about  their  positions  of  equilibrium.     (2)    The  dis- 
turbance is  propagated  at  a  uniform  rate,  each  displaced  particle 
transferring  its  motion  to  its  neighbors  by  pressure  or  through  some 
mechanical  connection.    The  moving  elements  of  the  medium 
possess  kinetic  energy  due  to  their  motion  and  potential  energy, 

173 


174  WAVE  MOTION 

due  to  their  displacements.  This  energy,  originally  derived  from 
the  source  of  disturbance,  is  passed  on  from  element  to  element, 
so  that  there  is  a  continuous  flow  of  energy  with  the  advancing 
waves. 

238.  Types  of  Waves. — The  displacements  in  the  case  of  water 
waves  do  not. extend  far  beneath  the  surface,  hence  disturbances 
are  propagated  in  two  dimensions  only,  in  superficial  waves. 
There  is  another  familiar  type,  resembling  water  waves  in  general 
shape,  which  may  be  propagated  along  a  linear  medium,  such  as  a 
wire  or  rope.  These  may  be  called  linear  waves  (although  the 
disturbance  extends  across  a  finite  area)  because  they  are  propa- 
gated in  one  direction  only.  Such  waves  may  be  studied  by 
filling  a  long  rubber  tube  with  shot  and  suspending 
it  from  a  tall  support,  holding  the  lower  end  taut  in 
the  hand.  If  the  tube  is  struck  a  sharp  blow  near 
the  lower  end,  a  distortion  resembling  a  wave  crest 
will  travel  slowly  to  the  upper  end,  where  it  will  be 
immediately  reflected  with  reversed  curvature,  on 
account  of  the  elastic  reaction  at  the  fixed  point. 
(Fig.  137,  a,  b,  c).  It  will  travel  to  the  lower  end 
and  be  reflected  back  and  forth  several  times  until 
its  energy  is  exhausted  by  friction.  This  is  a  solitary 
wave.  If  the  lower  end  is  rapidly  moved  back  and  forth  through 
a  small  amplitude,  by  properly  timing  the  displacements  a  series 
or  train  of  waves  of  opposite  curvatures  ("crests  and  hollows") 
will  travel  upward,  crossing  a  similar  train  reflected  downward. 
The  combined  effect  of  the  two  trains  is  to  cause  the  tube  to 
oscillate  between  the  positions  shown  by  the  full  and  the  dotted 
line  in  Fig.  137,  d. 

In  the  cases  mentioned  the  oscillations  of  the  medium  are  in 
part  or  altogether  at  right  angles  or  transverse  to  the  direction  of 
propagation,  so  that  the  displacements  of  the  boundary  of  the 
medium  give  rise  to  a  definite  wave  shape.  It  is  possible,  how- 
ever, for  the  vibrations  to  take  place  in  the  direction  of  propaga- 
tion of  the  wave,  as  is  the  case  with  one  component  of  the 
displacement  in  water  waves.  When  the  displacements  are 
altogether  in  the  direction  of  propagation  it  is  evident  that  the 
wave  can  have  no  shape,  as  the  boundaries  of  the  medium  are  not 
.displaced,  but  there  will  be  periodic  changes  in  density,  arising 


TYPES  OF  WAVES 


175 


from  the  fact  that  different  particles  are  at  any  instant  in  different 
phases  of  displacement,  so  that  in  one  region  they  will  be  crowded 
together,  while  in  another  they  will  be  separated.     This  may  be 
illustrated  by  a  row  of  massive  spheres,  connected  by  elastic 
cords  or  springs,  as  shown  in  Fig.  138,  a.     If  the  second  sphere 
were    immovable,    the    first    alone    would 
oscillate    when   pulled    downward    and   re- 
leased.    If  the  spheres  are  all  free  to  move, 
the  transmitted  impulse  will  set  all  in  vibra- 
tion.     On   account   of   the   inertia   of  the 
spheres  and  the  elasticity  of  the  connec- 
tions, the  displacement  of  each  sphere  will 
lag  behind  that  of  its  neighbor  below,  and 
each  vibration  will  be  in  a  different  phase, 
until  we  come  to  the  sphere  B,  which  be- 
gins its  first  vibration  when  A  begins  its 
second  vibration.     The  figure  shows  the  re- 
sultant effect  when  the  first  sphere  has  com- 
pleted one  vibration  (6)  and  one  and  a  half 
vibrations  (c)   after  it  first  moved  upward 
through  its  resting  point.     It  is  evident  from 
the  figure  that  the  conditions  of  condensation 
and  of  rarefaction  are  propagated  with  the 
velocity  of  the  wave.     There  is  no  change  of 
shape  in  the  system,  but  if  lines  proportional 
to  the  displacements  are  drawn  from  each 
resting  point,  to  the  right  for  upward  dis- 
placements, to  the  left  for  downward  dis- 
placements (that  is,  if  each  displacement  is 
rotated  through  90°  to  the  right  or  the  left) , 
a  smooth  curve  drawn  through  the  ends  of 
these  lines  will  have  the  general  shape  of  a 
transverse  wave  (6,  c).     We  have  thus  a  means  of  graphically 
representing  longitudinal  waves  in  a  way  clearly   coordinating 
them  with  transverse  waves. 

If  a  series  of  heavy  bars  are  attached  horizontally  at  equal 
intervals  to  a  suspended  wire,  and  if  the  lowest  bar  executes  tor- 
sional  vibrations,  waves  of  angular  displacement  will  travel  up 
the  wire.  Such  torsional  waves  may  be  represented  graphically 


6 
Fia.  138. 


176  WAVE  MOTION 

by  erecting  ordinates  proportional  to  the  angle  of  torsion  at  each 
point  on  an  axis  representing  the  wire. 

There  are  many  cases  where  wave  disturbances,  such  as  those 
of  sound  in  air,  are  propagated  in  three  dimensions  in  a  uniform 
medium.  These  disturbances  will  travel  equal  distances  in  all 
directions  in  equal  times,  hence  the  waves  will  be  spherical,  with 
the  source  as  a  center.  A  hemispherical  wave  of  this  type  would 
be  produced  in  a  block  of  rubber  by  striking  it  at  a  point. 

So  far  we  have  considered  the  effect  of  mechanical  disturbances  of  a 
medium  only.     The  idea  of  wave  motion  may,  however,  be  extended  to 
cases  where  any  physical  condition  in  a  medium  varies  periodically  at  each 
point  and  is  propagated  with  a  finite  velocity  through  the 
_  medium.     A  familiar  example  is  found  in  the  "  heat  waves  " 
which  travel  into  the  earth  as  a  result  of  the  periodic  heat- 
ing and  cooling  of  the  surface.     In  the  afternoon  the  sur- 
face reaches  a  maximum  temperature.     Owing  to  the  slow 
conduction  of  the  heat,  this  maximum  travels  slowly  down- 
ward, all  the  while  becoming  less  and  less,  owing  to  the 
fact  that  each  particle  passes  on  only  a  portion  of  the 
energy  received   by  it,   not  nearly  all,  as  in  the  case  of 
elastic  media.      At  night  the  surface  reaches  a  minimum 
Fio.  139.       temperature   which   penetrates  into   the  soil  at  the  same 
rate  as  the  maximum.     The  distribution  of  temperatures 
in  the  afternoon  and  at  night  are  represented  by  the  full  and  the  dotted 
line  in  Fig.  139.     The  abscissa  of  the  point  A  represents  the  average  tem- 
perature.    AB  is  the  distance  traveled  by  the  heat  wave  in  twenty-four 
hours.     Another  example  of  immaterial  waves  is  found  in  the  electrical 
waves  traveling  along  conductors  or  in  free  space,  due  to  periodic  change 
in  the  electrical  condition  at  different  points.     Light  waves  are  very  short 
electrical  waves  (§543). 

239.  Vibrations  in  Wave  Motion. — In  all  departments  of 
physics,  particularly  in  Sound,  Light,  and  Electricity,  waves  play 
an  important  part,  hence  the  study  of  wave  motion  is  of  funda- 
mental importance.  Since  periodic  displacements  or  changes  in 
condition  are  an  essential  feature  of  wave  motion,  it  is  necessary 
to  study  such  phenomena  in  detail.  The  only  periodic  motions 
which  lend  themselves  readily  to  simple  analysis  are  those  of 
uniform  motion  in  a  circle  or  the  projections  of  such  motions  along 
a  line,  the  latter  being  called  simple  harmonic  motions.  (§108  ek 


As  pointed  out  in  §111,  the  vibrations  of  all  elastic  bodies  must 


SIMPLE  HARMONIC  MOTIONS 


177 


Fia.  140. 


be  either  simple  harmonic  motions  or  compounded  of  such  motions 
(§248),  since,  for  small  displacements  at 
least,  the  force  of  restitution  is  propor- 
tional to  the  displacement. 

240.  Resolution  of  Simple  Harmonic 
Motions. — As  the  motion  is  a  linear  dis- 
placement, it  may  be  resolved  into  two 
or  more  components  like  any  other  dis- 
placement (§25).     If,  for  example,  the 

piston  rod  A B  (Fig.  140)  executes  simple  harmonic  vibrations  in  a 
horizontal  line  (the  projected  motion  of  the  crank  pin  on  a  fly- 
wheel), a  pin  P  attached  to  it  and  sliding  in  a  slotted  cross  bar 
attached  to  the  rod  CD  will  cause  the  latter  to  execute  a  simple 
harmonic  vibration  in  the  direction  of  its  length,  if  guides  allow  it 
to  move  only  in  that  direction.  If  the  amplitude  of  A  B  is  r,  the 
length  of  the  crank  arm,  that  of  CD  is  r  cos  a. 

241.  Superposition  of  Simple  Harmonic  Motions. — In  many 
cases  a  body  may  be  subjected  to  several  simultaneous  simple 
harmonic  displacements  in  the  same  or  in  different  directions  and 
of  the  same  or  different  periods.    Familiar  illustrations  are  found 
in  the  vibrations  of  musical  instruments  (§605  et  seq.)  and  when- 
ever different  sets  of  waves  are  superimposed  on  or  cross  each 
other.     If  the   displacements   are  entirely  independent,   it   is 
evident  that  the  resultant  effect  may  be  obtained  by  the  geomet- 
rical addition  of  displacements  (§13).     If  a  light  pendulum  is 

suspended  from  a  heavy  one,  as  shown  in  Fig.  141,  and 
both  set  in  vibration  in  the  same  plane,  at  a  given 
instant  the  total  displacement  of  the  lower  bob  is 
x  —  x1+x2;    or    if   the    pendulums    vibrate   at    right 
angles,  the  resultant  displacement  is  r=\/x*+y*.     In 
such  a  case  the  two  systems  are  not  entirely  inde- 
pendent, on  account  of  their  connections  and  inertia, 
and  the   two   displacements  will  not  remain  of  the 
simple  harmonic  type.     If  a  simple  pendulum  be  set 
in  vibration,  and  later  an  impulse  at  right  angles  to  its 
direction  of  motion  be  applied,  it  will  move  in  a  circular  or 
elliptic  orbit   (conical  pendulum),  or  in  a  line  inclined  to  its 
original  direction.     In  studying  these  effects  the  most  useful 
cases  to  consider  are  those  in  which  the  periods  of  the  com- 
ponents are  either  equal  or  in  some  simple  ratio  to  one  another. 
12 


Fia.  141. 


178 


WAVE  MOTION 


242.  Composition  of  Two  Simple  Harmonic  Motions  of  Same 
Period  and  in  Same  Line.  —  A  body  at  0  (Fig.  142)  has  a  horizontal 

simple  harmonic  motion  of  period 
T  and  amplitude  rr  When  the 
phase  of  this  vibration  is  e  a 
second  simple  harmonic  vibration 
of  the  same  period,  in  the  same  line, 
and  of  amplitude  r2  is  imparted  to 
the  body.  When  the  phase  of  the 
second  disturbance  becomes  cot  that 
of  the  first  is  wt  +  e;  e  is  the  phase 
difference.  To  find  the  resultant 
displacement  and  phase,  describe 
142.  circles  of  reference  of  radii  rl  and  r2 

about  0.     On  these  radii,  including 

the  angle  e,  complete  the  parallelogram,  and  draw  the  diagonal 
OC  =  R.    Then,  denoting  OP  by  xl9  OQ  by  x2,  and  OC  by  R, 


xl  =  rl  cos(wt  +  e) 

X2  ~  T1   COS   Wt 

and,  from  the  parallelogram  OC^CCj, 

&  =  rt2  +  r2a  +  2rxr2  cose 
If  x  is  the  resultant  displacement 


(1) 


(3) 


Hence  x  =  R  cos  (cot  +  6)  (4) 

This  holds  good  for  any  value  of  cot.  The  resultant  is,  therefore, 
a  simple  harmonic  motion  of  the  same  period  as  that  of  the  com- 
ponents ,  of  amplitude  R  and  with  a  phase  (cut  +  6)  intermediate 
between  the  phases  of  the  components.  It  is  the  projected  motion 
of  the  point  C  in  the  resultant  circle  of  reference  of  radius  R. 

The  above  results  may  also  be  obtained  from  the  component 
simple  harmonic  motions  (1)  and  (2)  without  use  of  the  circles  of 
reference.  For  the  sum  of  x^  and  x2  may  evidently  be  written  in 
the  form: 

x  =  (rt  cos  e  +  r2)  cos  a)t  —  (rt  sin  e)  sin  ajt. 

If  we  now  introduce  a  new  length  R  and  a  new  angle  0  such  that 
R  cos  6  =  (rl  cos  e+r2) 
R  sin  6  =  (rl  sin  e) 


SIMPLE  HARMONIC  MOTION 


179 


we  can  by  simple  trigonometry  obtain  (3)  and  (4)  and  also  an 
expression  for  tan  0. 

It  is  evident  from  (3)  that  R  is  a  maximum,  (r1  +  r2)t  when 
e  =  0  and  R  is  a  minimum,  (r^  —  rj,  when  e  =  180°. 

While  the  above  refers  to  the  addition  of  two  simple  harmonic 
motions  of  the  same  period,  we  can  extend  it  to  the  case  of  two 
vibrations  of  different  periods  by  supposing  the  phase  difference, 
e,  to  change  uniformly  with  the  time.  We  may  suppose  the  two 
motions  to  start  at  the  same  instant,  e  being  then  0.  At  time  t, 
the  value  of  e  will  be  (a)i  —  w2)t  =  2n(nl  —  n2)t,  where  nt  and  n, 
are  the  respective  frequencies.  When  (nl  —  n2)£  =  0,  1,  2,  3,  etc., 
cos  e  will  be  1  and,  from  (3),  R  will  be  a  maximum  (rl+r2). 
When  (rij  — n2)J  =  i,  |,  etc.,  R  will  be  a  minimum  (rx  — r2). 
The  interval  between  two  successive  maximum  values  of  R  is 
\l(nl  —  n^  and  the  number  of  maxima  per  second  is  (nl  —  n2). 
This  case  is  illustrated  by  "beats"  in  Sound  (§600). 

243.  Composition  of  Two  Simple  Harmonic  Motions  of  Same 
Period  at  Right  Angles. — If  the  amplitudes  of  the  respective 
vibrations  are  rx  and  r2,  construct  a  rectangle  with  sides  2rx  and 
2r2,  the  equilibrium  position  of  the  vibrating  particle  being  at  the 
center.  Construct  two  circles 
of  diameters  2rl  and  2r2,  as 
shown  in  Fig.  143.  The  pro- 
jections on  the  X  and  Y  axes 
respectively  of  points  moving 
uniformly  around  these  circles 
of  reference  will  give  the  x 
and  y  components  of  the  dis- 
placements of  the  body.  If 
the  former  is  in  advance  of 
the  latter  by  the  phase  angle 
e,  the  body  will  be  at  P  when 
the  y  displacement  begins. 
Divide  each  circle  into  the 
same  number  of  equal  parts,  Vlo  143 

beginning  at  Cl  and  C2,  and 

number  these  in  regular  order.  It  is  evident  that  the  successive 
positions  of  the  body  will  be  at  the  intersections  of  the  lines 
1-1,  2-2,  3-3,  etc.,  and  a  smooth  curve  drawn  through  these 


180 


WAVE  MOTION 


Fio.  144. 


points  will  give  the  orbit  of  the  body.     In  the  case  illustrated, 
3-7  where  e  =  45°,  this  path  is  an 

ellipse  inclined  to  the  axes. 
If  the  phase  difference  is  zero, 
2_g  the  path  is  a  straight  line,  the 
diagonal  BD.  If  e  =  90°,  the 
path  is  an  ellipse  with  vertical 
and  horizontal  axes,  or  a  circle 
if  r1=r2.  Orbits  correspond- 
ing to  different  values  of  e 
are  shown  in  the  top  row  of 
Fig.  145. 

If  the  periods  differ  slightly,  one 
vibration  will  gain  on  the  other  in 
phase,  and  the  orbit  will  run 
through  the  complete  cycle  of 
forms  shown  in  the  top  row  of  Fig.  145.  If  nv  nz  are  the  respective 
frequencies,  the  cycle  will  repeat  itself  whenever  one  component  gains  a 
whole  vibration  on  the  other,  or  nx— na  times  a  second. 

$  =  o          8=1/sT       S=1/4T      8 


1:1 


1:2 


1:8 


2:8 


FIQ.  145. 


244.  Composition  of  Two  Simple  Harmonic  Motions  at  Right 
Angles  with  Periods  in  Simple  Ratio. — Proceed  as  in  the  last  case, 


LISSAJOUS'  FIGURES  181 

but  divide  the  respective  circles  of  reference  into  a  number  of 
equal  parts  proportional  to  the  respective  periods,  so  that  the 
intervals  in  the  two  circles  will  be  traversed  in  equal  times.  Fig. 
144  illustrates  the  case  where  T1/T2  =  l:2f  and  the  angular  phase 
difference  is  45°  or  the  time  phase  difference  d  =  T2/S.  Orbits 
corresponding  to  other  phase  differences  and  to  the  ratios 
TJTi  =1:3  and  2:3  are  shown  in  Fig.  145. 

245.  Lissajous'  Figures. — Experimental  illustrations  of  these 
curves  were  first  obtained  by  Lissajous.     His  method  was  to 
reflect  a  beam  of  light  from  a  mirror  attached  to  one  end  of  a 
tuning  fork  to  a  corresponding  mirror  on  another  fork  vibrating 
in  a  plane  at  right  angles  to  the  first,  and  thence  on  a  screen.     The 
beam  is  displaced  by  both  forks,  and  the  spot  of  light  on  the  screen 
describes   the  resultant  path.     Another  method  is  to   use   a 
Y-pendulum,  as  shown  in  Fig.  146.     If  the  bob  vibrates  in  the 
plane  of  the  paper,  the  effective  length  is  PQ;  if  it  vibrates  at 
right  angles  to  this  plane,  it  is  CQ.     The  periods  in  the  two  planes 
will,   therefore,   be   different   and   independent.     By  properly 
adjusting  the  lengths  PQ  and  CQ  the  bob  may  be  made  to 
describe  the  various  Lissajous'  figures.     If  the  pendulum  has  a 
single  support,  Tl  —  T2f  and  the  bob  will  move  in  an  ellipse, 
circle,  or  straight  line,  according  to  the  differ- 
ence of  phase  between  two  impulses  given  to  it  _ 

at  right  angles. 

A  rectangular  rod  fastened  at  one  end  will 
vibrate  transversely  with  a  period  depending  on 
its  thickness.     If  the  diameters  parallel  to  two 
sides   are   different,   the  respective  periods  of 
vibration  will  be  inversely  as  the  diameters. 
If  drawn  aside  diagonally  and  released,  the  rod        FIO. 
will  not  continue  to  vibrate  in  that  direction, 
but  the  displacement  will  be  resolved  into  two  components 
parallel  to  the  diameters.     If  the  ratio  of  the  periods  is  simple, 
the  end  of  the  rod  will  describe  Lissajous'  figures. 

246.  Waves  due  to  Simple  Harmonic  Motion. — Consider  a  num- 
ber of  spheres  of  equal  masses  attached  to  each  other  by  elastic 
connections,  as  in  Fig.  147.     If  a  transverse  simple  harmonic 
vibration  is  imparted  to  the  first,  the  impulse  will  be  transmitted 
to  the  others  in  succession.     Suppose  the  phase  difference  between 


182 


WAVE  MOTION 


the  displacements  of  successive  spheres  to  be  one-eighth  of  a 
period.  When  a  has  completed  one  vibration,  6  has  completed 
seven-eighths  of  a  vibration,  etc.,  while  i  is  just  beginning  to 
move.  The  positions  of  the  spheres  will  be  at  the  projections 
on  the  vertical  line*  1,  2,  3,  etc.,  of  the  points  1,  2,  3,  etc.,  of  a 
circle  of  reference,  with  radius  equal  to  the  amplitude  of  the 
wave.  If  a  smooth  curve  be  drawn  through  these  positions,  it 
will  give  the  wave  form.  It  is  evident  that  the  abscissa  of  any 
point  on  this  curve  is  proportional  to  the  time  required  for  the 
disturbance  to  reach  that  point,  or  to  the  phase  angle,  and  its 
ordinate  to  the  sine  of  the  phase  angle  of  the  disturbance  at  the 
point.  Such  a  locus  is  called  a  harmonic  curve  or  sine  curve, 
and  gives  the  shape  of  a  transverse  wave  when  the  medium 
executes  simple  harmonic  vibrations.  If  the  particles  in  Fig.  137 


987654321 


Fio.  147. 

execute  simple  harmonic  vibrations,  the  longitudinal  wave  will 
be  of  the  same  type,  and  may  be  represented  by  a  sine  curve. 

The  period  and  the  amplitude  of  the  wave  are  the  same"  as 
those  of  the  simple  harmonic  motion  of  any  point  in  the  medium. 
The  wave  length  A  is  the  distance  between  any  two  consecutive 
points  in  the  same  phase  of  displacement,  for  example  a  and  i 
(Fig.  147).  If  v  is  the  velocity  of  propagation  of  the  wave, 
vT  =  ^,  since  X  is  the  distance  traversed  by  the  wave  during  a 
complete  vibration  of  the  "source,"  sphere  a.  If  n  =  l/T  is  the 
frequency  of  vibration,  v  =  rU,  the  length  of  the  train  of  waves 
sent  out  in  one  second. 

The  displacement  in  a  longitudinal  wave  presents  the  same 
aspect  if  looked  at  from  any  direction  in  a  plane  at  right  angles 
to  the  direction  of  propagation.  This  is  not  the  case  with  the 
transverse  waves  represented  in  Figs.  136, 147,  for  the  vibrations 


SIMPLE  HARMONIC  MOTION  183 

will  be  in  the  line  of  sight  if  viewed  in  the  plane  of  vibration, 
and  at  right  angles  to  the  line  of  sight  if  viewed  normally  to  this 
plane.  These  transverse  waves  have  a  sort  of  polarity,  there- 
fore, and  are  said  to  be  plane  polarized. 

Transverse  waves  may  be  set  up  in  a  cord  or  longitudinal 
waves  in  a  spiral  spring  by  fix- 
ing one  end  and  attaching  the 
other  to  a  vibrating  tuning  fork. 
The  amplitude  of  the  waves 
in  such  cases  may  be  much  FIQ.  148. 

greater  than  that  of  the  fork. 

If  a  beam  of  light  be  reflected  from  a  mirror  attached  to  the 
end  of  a  vibrating  fork,  and  again  reflected  to  a  screen  from  a 
revolving  mirror,  the  harmonic  curve  will  be  traced  on  the 
screen  by  the  spot  of  light.  Persistence  of  vision  will  cause  the 
path  to  appear  continuous. 

A  permanent  record  of  such  curves  may  be  made  by  causing 
a  bristle  attached  to  the  end  of  a  tuning  fork  to  trace  its  path 
on  the  smoked  surface  of  a  piece  of  glass  which  is  moved  past 
the  fork  at  a  uniform  rate  v. 

The  coSrdinates  at  the  time  /  of  a  point  P  on  the  sine  curve,  with  respect 
to  the  origin  0,  are  evidently  (Fig.  148). 


Eliminating  *, 

.     2nx        .    27: 
y-r  sm  ^--rsmy  x 

This  is  the  equation  of  a  sine  curve  repeating  itself  at  intervals  of  x***X. 

If  3/=»rsin  2xt/T  is  the  harmonic  displacement  at  a  given  point,  the 
disturbance  will  reach  a  point  at  a  distance  x  in  the  time  ti^x/v;  the 
disturbance  at  the  point  x  at  the  time  /  will  have  the  phase 


and 


This  is  the  equation  of  wave  motion.  At  a  given  time  /,  say  /  ««0,  it  gives 
the  instantaneous  picture  of  the  wave  train  as  a  sine  curve.  At  a  given 
point,  say  ar—0,  it  represents  the  simple  harmonic  vibration  of  the  medium 
at  that  point. 


184 


WAVE  MOTION 


247.  Superposition  and  Interference  of  Waves. — If  two  or  more 
trains  of  waves  are  superimposed,  each  will  give  rise  to  independ- 
ent displacements  of  the  medium.  The  resultant  effect  may  be 
obtained,  therefore,  by  plotting  each  train  of  waves  on  the  same 
axis,  with  relative  displacements  corresponding  to  their  phase 
differences,  and  adding  the  ordinates.  It  is  convenient  to  ex- 
press the  phase  differences  in  terms  of  wave-length.  If,  for  ex- 
ample, one  wave  starts  half  a  period  later  than  another,  it  should 
be  plotted  with  its  front  half  a  wave-length  behind  that  of  the 
first.  In  Fig.  149,  A,  B,  C,  the  full  line  represents  the  resultant 
of  two  waves  of  the  same  length  and  with  phase  differences  of  0, 
A/4,  and  A/2,  respectively.  In  the  last  case  the  resultant  effect 


Fio.  150. 

is  zero  if  the  amplitudes  are  equal.  The  modification  of  ampli- 
tude due  to  the  superposition  of  waves  is  called  interference.  It 
is  evident  that  the  length  of  the  resultant  wave  is  the  same  as 
that  of  its  components,  and  that  it  is  a  harmonic  curve  if  they  are 
harmonic  curves. 

248.  Complex  Waves. — Waves  of  different  lengths  may  be  com- 
bined in  the  same  manner.  If  the  lengths  are  in  simple  ratio  to 
one  another,  all  the  resultant  waves  in  a  train  will  be  of  the  same 
form,  but  this  form  will  vary  with  the  phase  difference,  and  will 
not  be  a  sine  curve.  This  is  illustrated  by  Fig.  150,  A,  B,  C, 
which  shows  the  resultant  of  two  waves  of  lengths  in  the  ratio 
1:2,  and  having  different  phase  relations.  Fig.  151  illustrates 


COMPLEX  WAVES  185 

the  case  where  the  lengths  are  as  1 : 3  and  the  phase  difference 
zero. 

If  the  components  have  lengths  which  are  not  in  simple  ratio, 
successive  waves  will  not  be  of  the  same  shape,  as  the  length  of 
the  longest  wave  will  not  be  a  common  multiple  of  the  lengths  of 
the  component  waves.  If  there  are  only  two  components, 
however,  with  frequencies  n4  and  n2, 
one  wave  will  gain  its  own  length 
on  the  other  in  l/Cr^— n2)  second, 
and  the  wave  train  will  consist  of 
similar  groups  repeating  themselves 
nt  —  n2  times  a  second.  The  length 
of  each  group  will  be  the  least  com-  FIQ.  151. 

mon  multiple  of  the  lengths  of  the 

components.  Fig.  152  shows  the  effect  of  superimposing  two 
trains  of  waves  of  lengths  having  the  ratio  3  :4.  The  graphical 
representation  of  "beat"  waves  in  sound  would  resemble  this 
figure  (§602).  Such  forms  may  be  obtained  experimentally  by 
the  optical  method  for  obtaining  sine  curves  described  in  §246, 
the  beam  of  light  being  reflected  successively  from  two  forks 
vibrating  in  the  same  plane,  and  giving  beats,  and  from  a  rotating 
mirror  to  a  screen. 


Fio.  162. 

The  displacements  of  the  medium  may  be  the  resultant  of  two 
displacements  at  right  angles.  The  example  of  water  waves  has 
already  been  mentioned.  If  one  end  of  a  cord  be  attached  to  the 
end  of  a  rectangular  rod  vibrating  transversely  parallel  to  each  of 
its  diameters,  the  end  of  the  rod  will  describe  Lissajous'  figures 
(§245)  and  each  element  of  the  cord  will  do  the  same.  If  the  two 
diameters  of  the  rod  are  equal,  each  element  of  the  cord  will 
move  in  a  circle  or  ellipse  in  a  plane  transverse  to  its  lengh, 


186  WAVE  MOTION 

but  the  phases  will  differ  from  point  to  point,  so  that  at  a  given 
instant  the  cord  will  have  the  shape  of  a  corkscrew.  Such  a 
wave  is  said  to  be  circularly  or  elliptically  polarized. 

249.  Fourier's   Theorem. — The   illustrations  given   show   that    various 
complicated  forms  may  be  obtained  by  the  addition  of  simple  harmonic 

waves  of  different  lengths  and  phases,  and  that  these  waves 
S/~\       will  be  of  persistent  form  if  the  periods  of  the  components 
j        are  simple  fractions  of  the  periods  of  the  longest  component. 
Fourier  proved  that  any  periodic  disturbance  or  wave  form 
of  permanent  type  could  be  represented  as  the  summation 
^     of  a  number  of  simple  harmonic  terms  of  the  form 

X^TV  sin  tot  +  rt  sin  2(at  +  rt  sin  3^-1-  •  •  •,  etc., 
the  periods  and  wave-lengths  of  the  components  having 
the    ratios  1,   £,    £,   i,   etc.      Fig.    151    shows   that   the 
Fia.  153.          resultant  is  approaching  a  rectangular  form,  which  may 

be  finally  attained  by  adding  shorter  waves. 

The  forms  of  complex  waves  may  be  projected  by  the  following  device 
(Fig.  153) :  A  screen  with  a  slit  opening  at  0  is  placed  in  front  of  a  horizontal 
stretched  wire  AB,  which  is  illuminated  by  the  lens  L.  An  image  of  the 
segment  opposite  the  opening  may  be  thrown  on  a  screen  S  after  reflection 
from  a  rotating  mirror  M .  If  the  wire  is  at  rest  the  image  of  the  illuminated 
segment  will  be  drawn  out  in  a  dark  straight  line  on  the  screen.  If  the 
wire  vibrates  in  a  vertical  plane  the  images  of  the  segment  in  its  successive 
phases  as  the  wave  passes  0  will  be  laid  off  end  to  end  on  the  screen,  giving 
the  actual  form  of  the  wave  passing  the  opening. 

250.  Velocity  of  a  Wave  on  a  Cord. — Let  the  wave  be  supposed 
to  be  moving  toward  the  left  with  a  velocity  v.     It  will  simplify 
the  problem  without  essentially  changing  it,  if  we  now  suppose 
the  cord  to  be  given  a  velocity  v  toward  the  right.     The  wave 
will  then  stand  still  and  every  part  of  the  cord,  as  it  comes  to 
the  wave,  will  pass  through  it  with  velocity  v.    This,  in  fact,  is 
what  may  often  be  noticed  in  the  use  of  a  chain  hoist.     If  the 
chain  be  started  in  rapid  motion  (there  being  no  load  on  the 
lower  pulley),  a  bend  impressed  on  the  chain  will  sometimes 
remain  stationary  for  a  short  time,  and,  if  the  chain  be  suddenly 
arrested,  the  bend  will  move  off  in  the  opposite  direction  with 
(approximately)  the  speed  which  the  chain  had.     The  relative 
velocity  depends  only  on  the  mass  and  tension  of  the  chain. 

Now  let  QR  be  a  small  part  of  the  wave,  its  length  I  being  so 
short  that  it  may  be  regarded  as  an  arc  of  a  circle  (the  circle  of 
curvature).  Draw  tangents  TQ  and  TR  and  complete  the  par- 


VELOCITY  OF  A  WAVE  ON  A  CORD 


187 


allelogram  QTRS.  The  velocities  at  Q  and  R  may  be  represented 
by  QT  and  TR.  As  each  part  of  the  cord  passes  from  Q  to  R  in 
time  t  it  will  have  an  acceleration,  a,  toward  the  center  of  curva- 
ture, that  is,  in  the  direction  of  the  diagonal  TS.  Since  TS  repre- 
sents a  velocity  which,  added  to  QT,  gives  QS  or  TR,  it  represents 
the  change  of  velocity,  at,  in  the  time  t.  Hence 

TS    at 


The  only  forces  that  act  on  the  part  QR  of  the  cord  are  the 
equal  forces,  F,  at  its  ends  due  to  the  tension  in  the  cord.  These 
may  be  represented  by  TQ  and  TR  and  their  resultant,  repre- 
sented by  TS,  is  the  force  that  causes  the  central  acceleration 


of  the  part  QR  of  the  cord.     If  the  mass  of  unit  length  of  the 
cord  is  m,  the  mass  of  the  part  QR  is  ml.     Hence 

TS^mla 
QT~  F 

Equating  the  above  values  of  TS/QT  and  noting  that  l  =  vtt  we 
get  v 

(It  is  evident  that  belting  traveling  with  this  velocity  will  exert 
no  pressure  on  a  pulley.     See  §47.) 

251.  Velocity  of  Elastic  Waves. — It  might  be  expected  that  the  velocity 
of  waves  in  an  elastic  medium  would  depend  upon  the  elasticity,  which 
determines  the  rate  at  which  an  impulse  is  transmitted  from  one  element 
to  another  (in  a  perfectly  rigid  and  incompressible  medium  the  effect  would 
be  instantaneous),  and  the  density,  which  exercises  a  retarding  influence, 
on  account  of  the  inertia  of  the  displaced  elements.  The  derivation  of  the 
exact  relation  between  the  velocity,  the  density  p,  and  the  coefficient  of 
elasticity  E  is  in  some  cases  mathematically  difficult,  but  the  general 


188 


WAVE  MOTION 


form,  at  least,  is  readily  obtained  from  a  consideration  of  the  dimensions 
of  the  quantities  involved  (§154).  If  the  velocity  depends  solely  on  p 
and  E,  we  may  write  v  —  kE*py,  where  x  and  y  are  unknown  powers,  and 
fc  a  factor  of  proportionality.  Substituting  dimensional  expressions  for 
the  quantities  (remembering  that  E  is  force  per  unit  area  and  p  is  mass  per 
unit  volume),  we  have 


(T) 


/M\ 
\L3/ 


.C 


By  inspection  we  find  with  respect  to  T  that  x  must  be  J.    To  make 
M  disappear  from  the  right-hand  side,  y  must  be  —  J.     Therefore 

•-s/f 

The  exact  relation  may  be  easily  found  in  some  simple  cases.     Suppose 
the  front  of  the  disturbance  in  a  longitudinal  wave  in  a  medium  of  unit 
_l__  cross-section  to  be  at  A  (Fig.  155)  at  one  instant 

and  at  B  a  short  time  t  later.  The  velocity  of  the 
wave  is,  therefore,  v-l/t,  where  l~*AB.  An  im- 
aginary plane  A  in  the  medium  is  displaced  to 
D,  a  distance  x,  by  compression.  If  I  is  very 
small,  the  density  of  the  substance  is  practically 
uniform  between  D  and  B,  and  the  center  of  mass 
of  the  element  is  displaced  from  C  to  C",  a  dis- 

tance x/2.  The  average  velocity  of  the  center  of  mass  is  x/2t  and  its  final 
velocity  x/t.  The  final  force  acting  on  the  element  is  Ex/I,  where  E  is 
Young's  modulus  in  the  case  of  a  solid,  or  the  modulus  of  elasticity  of 
volume  in  case  of  a  gas.  The  average  force  is  half  of  the  above.  Equating 
the  work  done  by  this  force  to  the  acquired  kinetic  energy  due  to  the  motion 
of  the  center  of  mass,  we  have 


\B 


Fio.  155. 


Ex 


therefore,  7=v»» 


E 


This  applies  to  longitudinal  waves  in  a  wire  or  rod  or  to  sound  waves  to 
any  medium. 

262.  Reflection  of  Waves. — When  a  transverse  wave  reaches 
the  fixed  end  of  a  cord,  the  displacement  is  immediately  reversed 
in  direction  by  the  elastic  re- 
action of  the  fixed  end.  The 
wave  is,  therefore,  reflected  ^~^X^_^X^ 


Fio.  156. 


with   reversal   of   phase   of 

displacement,  as  shown   in 

Fig.    156.      Apparently   the    incident    wave    has    disappeared 

through  the  end,  while  a  wave  of  opposite  displacement  has 

entered,   traveling   in   the   opposite   direction,    and    at  every 


REFLECTION  OF  WAVES  189 

instant  exactly  neutralizing  the  displacement  of  the  end  which 
would  be  caused  by  the  incident  wave  if  the  end  were  free.  When 
a  continuous  train  is  reflected,  the  effect  is  as  though  a  train  of  in- 
definite length  had  been  cut  in  two  when  a  wave-front  reaches  A, 
the  fixed  point  (Fig.  156),  and  the  waves  to  the  right  immediately 
reversed  in  direction,  while  the  incident  waves  continue  their 
motion  unchanged;  or  as  though  a  train  of  incident  waves  were 
traveling  through  a  mirror,  while  their  inverted  images  proceed 
out  of  it  in  the  opposite  direction. 

If  one  end  of  the  cord  is  free,  when  the  wave  reaches  that 
point,  the  end,  having  nothing 
beyond  to  restrain  it,  has  an 
outward   displacement  twice 
as  great  as  though  the  cord  FIQ.  157. 

were  continuous,  and  it  will, 

therefore,  immediately  start  a  wave  of  the  same  phase  in  the  re- 
verse direction.  After  half  a  period  of  vibration  it  will  return 
through  the  resting  point  in  the  opposite  direction,  and  will 
start  a  backward  wave  with  phase  opposite  to  that  of  the  in- 
cident wave.  It  is  as  though  a  train  has  been  cut  in  two  when 
a  crest  is  at  the  free  end  B  (Fig.  157),  and  the  right-hand  sec- 
tion immediately  reversed;  or  as  if  an  advancing  train  were 
passing  through  a  mirror  while  its  erect  image  emerged  from  it. 

To  show  in  another  way  the  difference  between  reflection  at  a 

free  end  and  at  a  fixed  end, 
^..^"""^  suppose  that  the  part  of  the 
direct    wave-train   that   first 
FIG.  168.  reaches  B  and  begins  to  be 

reflected  is  as  represented  in 

Fig.  158.  If  we  compare  this  with  Fig.  156,  it  is  seen  that  in  re- 
flection at  a  free  end,  as  compared  with  that  at  a  fixed  end,  there 
is  a  delay  of  half  a  period  in  the  reflection  of  the  wave  of  oppo- 
site phase,  as  though  the  right-hand  section  in  Fig.  156  were  held 
at  rest  for  half  a  period  before  starting  in  the  negative  direction. 

Reflection  of  longitudinal  waves  may  be  illustrated  by  the  con- 
duct of  a  row  of  elastic  pendulums  of  the  same  size,  as  shown  in  Fig. 
159a,  the  last  resting  against  a  fixed  obstacle.  If  a  is  drawn 
aside  and  released,  it  will  impart  an  impulse  to  b,  this  in  turn 
to  c,  etc.,  and  a  compression  wave  will  travel  to  the  other  end 


B 


I 


190  WAVE  MOTION 

of  the  row;  g  cannot  move,  but  will  be  compressed,  and 
through  its  elastic  reaction  it  will  almost  immediately  start  a 
compression  wave  in  the  opposite  direction.  When  this  wave 
reaches  the  free  end,  a  will  fly  out  without  restraint,  leaving  a 
rarefaction  behind  it;  or,  if  elastically  connected  with  6,  it  will 
at  once  send  back  a  rarefaction  wave.  In  any  event,  after  ex- 
ecuting half  a  vibration  it  will  swing  back  through  its  equilibrium 
position  and  reflect  a  compression  wave  to  the  right. 

If  there  are  two  rows  of  elastic  pendulums  of  different  masses, 

as  in  Fig.  1596,  displacements  will  be 

immediately  transmitted  across  At  no 
matter  in  which  direction  the  wave  is 
moving,  but  the  wave  will  be  also 
partially  reflected.  If  the  wave  travels 
to  the  right,  reflection  of  a  at  A  will  be 
immediate;  if  it  travels  to  the  left,  the 
more  massive  sphere  6  will  continue 
FIO.  159.  after  impact  to  move  to  the  left,  and 

will  return  through  its  resting  point, 

to  send  a  wave  back  to  the  right,  at  the  expiration  of  half  its 
period  of  vibration. 

The  cases  mentioned  illustrate  the  general  principle  that  the 
displacements  in  a  medium  have  a  minimum  amplitude  at  a 
fixed  or  constrained  boundary;  a  maximum  amplitude  at  a  free 
boundary  or  one  with  diminished  constraint.  Important  illus- 
trations of  this  principle  occur  in  cases  where  waves  pass  from 
a  light  to  a  dense  medium  or  vice  versa  (§§609,  686). 

253.  Stationary  Waves. — Consider  a  train  of  waves  in  a  cord 
moving  to  the  right,  while  a  similar  train  (reflected  or  independ- 
ent) moves  to  the  left.  Interference  will  take  place,  and  the 
resultant  displacement  of  the  medium  at  a  given  point  and  time 
will  be  the  sum  of  the  individual  displacements.  Plot  the 
positions  of  the  waves  at  successive  instants  (say  at  intervals 
of  an  eighth  of  a  period).  If  the  incident  train  is  represented 
by  a  light  line,  the  reflected  train  by  a  dotted  line,  and  the 
resultant  by  a  heavy  line  (Fig.  160),  it  will  be  seen  that  there 
are  always  points  of  zero  displacements  N  (or  of  minimum  dis- 
placement if  the  amplitudes  are  unequal)  at  intervals  of  half 
a  wave-length,  where  the  waves  always  meet  in  opposite  phases. 


WAVES  IN  A  LIQUID  191 

Half  way  between  these  points,  at  L,  the  waves  will  always 
meet  in  the  same  phase,  and  the  displacement  will  be  a  maxi- 
mum. The  former  positions  are  called  nodes,  the  latter  loops 
or  antinodes.  Between  the  nodes  the  medium  oscillates  back 
and  forth,  the  direction  of  the  displacements  being  opposite 
in  adjacent  segments,  so  that  at  any  instant  the  cord  has  a 
more  or  less  sinuous  shape, 
except  at  intervals  of  half  a 
period,  when  it  passes  through 
the  undisturbed  straight  posi- 
tion (Fig.  137d).  The  same 
conclusions  apply  to  longi- 
tudinal waves.  Disturbances 
of  this  sort  are  called  sta- 
tionary waves.  It  is  evident 
that  when  these  arise  from  the 
interference  of  incident  and  reflected  waves  there  must  be  a 
node  at  a  fixed  or  constrained  boundary,  a  loop  at  a  free  or 
unconstrained  boundary. 

Fig.  161  is  the  graphical  representation  of  stationary  waves  of 
longitudinal  type.  The  displacements  have  just  begun  to  return 
from  maximum  elongation,  from  the  full  to  the  dotted  line. 
This  indicates  that  the  particles  to  the  left  of  Nl  and  those  to  the 
right  of  N2  are  moving  in  the  negative  direction,  while  those 
between  Nt  and  N2  are  moving  in  the  positive  direction.  Con- 
sequently the  particles  on  opposite  sides  of  N2  are  approaching 
that  point,  while  those  on  opposite  sides  of  Nt  are  receding 
from  it.  At  N2  there  will  be  a  condensation,  at  Nl  a  rarefaction. 

After  half  a  period  conditions 
Lj          ^     ^^       will    be    reversed.      In    the 

neighborhood  of  Llf  L2,  how- 
161  ever,  the  particles  are  moving 

in  the  same  direction  with  ap- 
proximately the  same  velocity,  so  that  their  relative  positions 
are  only  slightly  changed.  It  follows  that  at  the  nodes  there  are 
the  greatest  variations  of  pressure,  and  the  least  motion;  at  the  loops, 
the  smallest  variations  of  pressure  and  the  greatest  motion. 

254.  Waves  in  a  Liquid.— Some  of  the  most  interesting  proper- 
ties of  wave  motion  may  be  illustrated  by  waves  on  the  surface  of 


192  WAVE  MOTION 

a  liquid,  such  as  water.  The  initial  displacement  may  arise  from 
differences  of  level  caused  by  some  external  force,  such  as  the 
impact  of  a  pebble,  winds,  etc.  The  effect  of  gravity,  of  fluid 
pressure,  and  of  surface  tension  is  to  restore  the  original  level, 
but,  on  account  of  their  inertia,  the  particles  are  displaced  beyond 
their  equilibrium  positions,  just  as  in  the  case  of  vibrations  of  a 
liquid  in  a  U-tube.  Horizontal  as  well  as  vertical  displacements 
must  occur,  as  in  the  case  of  the  liquid  in  the  bend  of  the  U-tube. 
There  is,  therefore,  a  longitudinal  as  well  as  a  transverse  compo- 
nent. These  displacements  are  simple  harmonic,  because  the 
resultant  pressure  on  an  element  is  proportional  to  its  vertical 
displacement  from  the  undisturbed  surface.  We  have  seen 
(§236)  that  on  a  crest  the  element  moves  forward,  in  the  hollow 
backward,  in  intermediate  positions  both  vertically  and  hori- 
zontally. Fig.  162  shows  the  positions  and  directions  of  rotation 
abcdefgh  i 


Fio.  162. 

of  a  number  of  particles  originally  at  rest  on  the  surface  in  the 
positions  under  a,  6,  c,  etc.,  the  phase  difference  between  succes- 
sive displacements  being  an  eighth  of  a  period.  Particle  a  is 
subject  solely  to  a  downward  acceleration,  particle  e  to  an 
upward  acceleration;  particles  c  and  g  are  subject  solely  to 
horizontal  accelerations,  due  to  the  lateral  pressure,  as  they 
are  in  the  horizontal  plane  of  equilibrium.  We  thus  find  that 
there  is  a  difference  of  phase  of  a  quarter  period  between  the 
vertical  and  horizontal  accelerations,  in  accordance  with  the  ob- 
served fact  that  the  disturbed  elements  move  in  elliptic  or  circular 
orbits.  It  is  evident  that  the  wave  form  is  not  a  sine  curve. 

The  expression  for  the  velocity  of  liquid  waves  is  complicated 
and  cannot  be  derived  here.  It  is  sufficient  to  say  that  large 
waves  are  maintained  by  gravity  alone,  and  that  the  velocity  is 
independent  of  the  density  of  the  liquid,  as  the  force  acting  is 
proportional  to  the  weight  of  the  displaced  elements,  and  hence 
will  produce  the  same  acceleration,  whatever  the  density.  The 
velocity  increases  with  the  wave-length,  so  that  one  may  fre- 
quently see  a  train  of  long  water  waves  sweeping  through  a  train 


REFRACTION  OF  WAVES 


193 


of  shorter  waves  and  leaving  them  behind.  When  the  liquid  is 
shallow,  the  velocity  diminishes  with  the  depth.  The  very  small 
waves  are  maintained  by  surface  tension  alone,  so  that  they  are 
analogous  to  transverse  waves  in  an  elastic  membrane.  In  the 
case  of  these  waves  the  velocity  increases  as  the  wave-length 
diminishes,  and  is  also  dependent  upon  the 
density  and  the  surface  tension.  Such  waves 
are  called  ripples. 

255.  Refraction   of   Waves. — Waves   move 
more  slowly  in  shallow  than  in  deep  water. 
Hence  if  the  front  AB  of  an  ocean  wave  mov- 
ing in  the  direction  of  the  arrow  (Fig.  163)  ap- 
proaches a  beach  CD,  the  nearer  end,  B,  of 
the  wave  will  be  retarded  more  than  A,  being 

in  shallower  water.  The  wave  front  will  swing  around  into  the 
successive  positions  A'B'  and  A"B" ',  and  will  finally  become 
parallel  to  the  shore  line.  This  change  in  direction  due  to 
change  in  velocity  is  called  refraction.  Similar  effects  are,  we 
shall  see,  shown  by  other  waves,  such  as  sound  and  light,  when 
they  pass  from  one  medium  to  another  in  which  they  travel  with 
different  velocity. 

256.  Propagation  and  Reflection   of    Ripples. — Experiments 
with  ripple  waves  may  be  shown  by  the  following  arrangement. 
A  shallow  wooden  box  with  a  glass  bottom,  about  two  feet  square, 
is  mounted  on  legs  like  a  table,  carefully  leveled,  and  partly 
filled  with  water.     Light  may  be  projected  upward  through  the 
bottom  from  an  arc  light  placed  beneath  the  box,  or  by  reflecting 

-  a  divergent  beam  of  sunlight  upward  by 

.    '  an  inclined  mirror.     Ripples  on  the  sur- 

face will  by  their  lens  effect  change  the 
distribution  of  light  on  the  ceiling  so  that 
02'  the  motion  of  each  ripple  may  be  followed. 
If  the  middle  of  the  surface  is  touched 
with  a  nail,  a  circular  ripple  will  diverge 
from  that  point.  If  the  surface  were 
larger,  this  wave  would  at  a  later  time 
occupy  the  position  of  the  circle  (Fig. 
164),  but  it  will  be  in  part  reflected  from  the  four  sides.  The 
reflected  segments  are  exactly  like  the  missing  segments  of  the 

13 


•ca 

Fia.  164. 


194 


WAVE  MOTION 


FIG.  165. 


outgoing  wave,  reversed  in  direction.  These  reflected  waves  have 
centers  at  Clt  C2,  C3  and  C4,  the  "images"  of  the  source  C,  which 
are  evidently  at  the  same  distance  from  the  walls  as  the  source 
itself,  since  C  and  the  other  centers  of  curvature  are  symmetric- 
ally situated  with  respect  to  the  walls. 
These  reflected  waves  will  cross  each  other 
and  be  subject  to  repeated  reflections 
("multiple  reflection"),  their  curvature 
all  the  while  decreasing,  until  we  have  a 
rectangular  system  of  straight  ripples. 

If  a  circular  wave  strikes  a  bent  sheet 
of  metal  of  the  same  curvature  as  the 
wave,  the  latter  will  be  reflected  without 
change  of  curvature,  converge  to  its  start- 
ing point,  and  diverge  from  it  on  the  opposite  side  (Fig.  165a). 
If  the  strip  has  a  greater  curvature  than  the  wave,  the  edges 
of  the  latter  will  be  first  reflected,  so  that  its  curvature  is 
increased  (6).  It  will  converge  to  C",  a  "real  image"  or 
"  focus."  If  the  strip  is  concave  with  less  curvature  than  the  wave 
(c)  the  latter  may  be  made  divergent,  with  a  "virtual"  image  at 
C'.  If  the  strip  is  convex  toward  the  wave,  the  reflected  wave 
will  always  diverge  from  a  virtual  center  behind  the  mirror  (d.) 
If  the  surface  is  touched  with  a  long  straight  strip  of  metal 
a  straight  ripple  will  be  produced.  If  this  strikes  a  screen  with 
a  small  slit  in  it  (Fig.  166e)  the  disturbance  will  pass  through 
this  hole  and  set  up  a  semicircular  wave  on  the  other  side. 
The  remainder  of  the  wave  will  be  reflected  as  a  straight  line. 
If  a  number  of  nails  are  driven  at  equal  distances  through  a 
strip  of  wood  and  dipped  into  the  water,  circular  waves  will 
diverge  from  the  points  of  contact.  At  a  little  distance  these 
wavelets  will  blend  into  a  straight 
ripple  corresponding  to  their 
common  tangent  (/).  At  other 
points  the  ripples  cross  each 
other  in  all  phases,  and  their 
effect  will  vanish  because  of  in- 
terference. We  may,  therefore, 

consider  that  a  linear  wave  front  is  due  either  to  a  continuous 
linear  disturbance  or  to  a  number  of  neighboring  point  dis- 


Fio.  166. 


REFRACTION  OF  RIPPLES 


195 


turbances,  each  sending  out  circular  waves.  In  (e)  for  example 
only  the  point  in  the  opening  is  effective  for  transmission.  The 
latter  conception  is  often  useful  (§639). 

If  a  screen  S  projects  part  way  across  the  tank  (g),  the  portion 
AS  of  an  incident  wave  will  be  reflected;  the  remainder  SB  of  the 
wave  will  pass  the  screen.  It  will  be  noted  that  the  end  of  the 
transmitted  wave  front  will  bend  into  the  shadow  of  the  screen, 
and  the  end  of  the  reflected  wave  will  bend  into  the  region 
formerly  occupied  exclusively  by  the  other  half  of  the  wave.  S 
is  apparently  a  center  of  disturbance  for  both  these  waves.  This 
effect  is  called  diffraction.  By  noting  the  resemblance  of  the  ends 
of  the  waves  in  this  case  to  those  in  the  preceding  case  (/)  the 
explanation  will  be  made  clear. 

267.  Refraction  of  Ripples. — Advantage  may  be  taken  of  the 
fact  that  the  velocity  of  water  waves  diminishes  with  the  depth  to 
illustrate  refraction.  On  the  bottom  of  the  tank  lay  a  piece  of 
thick  glass,  so  that  the  water  over  it  is  about  one-fourth  as  deep 
as  elsewhere.  A  linear  ripple  is  started  by  touching  the  surface 
with  a  strip  of  metal.  On  reaching  the  edge  of  the  glass  plate 


" 


Fia.  167. 


Fio.  168. 


the  end  B  is  retarded  and1  the  wave  will  swing  into  the  position 
A'B',  as  in  Fig.  163. 

If  the  incident  wave  is  circular,  the  middle  will  be  more  re- 
tarded than  the  edges  if  the  wave  comes  from  the  deeper  water, 
and  the  curvature  of  the  wave  will  be  diminished  (Fig.  lQ7h). 
If  the  wave  travels  from  the  shallow  region,  the  contrary  will  be 
the  case  (i).  The  centers  of  curvature  or  "images"  of  the  source 
will  be  at  C'  (outside  the  tank  in  h). 

If  a  prismatic  sheet  of  glass  is  laid  on  the  bottom  (/)  a  linear 
wave  front  AB  will  be  rotated  both  in  approaching  and  leaving, 
and  the  final  direction  will  be  A"B".  If  pieces  of  glass  with 
convex  or  with  concave  edges,  like  sections  of  lenses,  are  laid  on 
the  bottom,  the  center  of  a  passing  circular  wave  will  be  more 


196  WAVE  MOTION 

retarded  than  the  edges  in  the  first  case  (fc),  less  retarded  in  the 
second  (I),  resulting  in  changes  of  curvature.  The  "images"  of 
the  source  will  be  at  C". 

268.  Interference  of  Ripples. — If  two  nails  simultaneously  touch  the 
water  at  different  points  two  circular  waves  will  be  set  up,  which  will  cross 
and  interfere  with  each  other.  They  pass  so  quickly,  however,  that  it  is 
difficult  to  observe  them.  Better  results  will  be  secured  if  a  continuous 
series  of  waves  can  be  produced,  and  still  better  results  if  there  is  a  system 
of  stationary  waves.  A  very  satisfactory  method  of  securing  this  result  is 
to  put  mercury  in  a  circular  glass  dish  at  least  four  inches  in  diameter,  and 
maintain  periodic  disturbances  at  the  center  by  a  glass  fiber  attached  to 
the  vibrating  prong  of  a  tuning  fork  Continuous  trains  of  circular  ripples 
will  diverge  from  the  center,  while  reflected  circular  ripples  will  converge 


Fio.  169. 

toward  that  point  The  result  will  be  a  system  of  circular  stationary  waves, 
as  illustrated  in  Fig  169.  They  may  be  projected  on  a  screen  by  reflected 
light,  and  made  more  distinct  by  using  a  lens. 

If  two  glass  fibers  are  attached  to  the  fork  near  each  other,  two  trains  of 
waves  will  be  maintained,  and  each  will  form  its  own  system  of  stationary 
waves.  At  all  points  on  the  surface  where  the  outgoing  waves  meet  each 
other  in  the  same  phase  (that  is,  where  the  difference  of  the  respective 
distances  to  the  two  sources  is  zero  or  any  whole  number  of  wave  lengths), 
the  waves  will  reinforce  each  other.  In  regions  where  they  meet  in  opposite 
phases  (the  differences  of  path  being  some  odd  multiple  of  a  half  wave 
length),  they  will  destructively  interfere  with  each  other.  Along  certain 
lines,  therefore,  there  will  be  no  disturbance  by  either  outgoing  or  reflected 
ripples  (Fig.  482).  Between  these  lines  segments  of  the  stationary  waves 
appear,  as  shown  in  Fig.  169. 


ENERGY  AND  INTENSITY  OF  WAVES  197 

259.  Energy  and  Intensity  of  Waves.  —  The  energy  of  a  vibrat- 
ing body  is  proportional  to  the  square  of  its  amplitude 
(§61).  Each  vibrating  element  of  mass  in  a  medium  traversed 
by  waves  will,  therefore,  possess  energy  proportional  to  the 
square  of  its  amplitude,  and  this  energy  will  flow  forward  with 
the  advancing  waves.  The  intensity  of  waves  in  a  given  region  is 
defined  as  being  proportional  to  the  amount  of  energy  passing 
per  second  through  unit  area  at  right  angles  to  the  direction  of 
propagation;  hence  the  intensity  is  proportional  jointly  to  the 
square  of  the  amplitude  and  the  velocity  of  the  waves.  In  a 
viscous  medium,  such  as  molasses  or  lead,  they  rapidly  decay  in 
amplitude  and  disappear,  owing  to  the  absorption  of  energy  by 
internal  friction.  This  effect  is  known  as  damping.  Fig.  138 
represents  the  form  of  a  damped  train  of  waves.  If  there  is  no 
such  loss  the  same  quantity  of  energy  will  persist  .in  a  given 
wave,  no  matter  how  far  it  travels,  or  how  the  dimensions  and 
form  of  the  wave  front  may  change.  If  such  waves  travel  in  a 
wire  or  any  other  channel  of  constant  cross-section  the  intensity 
will  be  independent  of  the  distance  from  the  source,  as  the  wave 
front  will  remain  of  constant  area.  This  is  illustrated  by  the 
transmission  of  sound  waves  through  a  speaking  tube  or  of  light 
waves  in  a  parallel  beam.  In  the  case  of  circular  waves  on  a 
surface,  a  constant  amount  of  energy  will  remain  in  a  wave  of 
circumference  which  increases  directly  as  the  distance  from  the 
source;  hence  the  intensity  must  vary  inversely  as  the  distance, 
and  the  amplitude  inversely  as  the  square  root  of  the  distance. 
In  the  case  of  spherical  waves,  the  energy  will  remain  constant 
within  a  spherical  shell  of  the  thickness  of  one  wave  length 
and  with  surface  increasing  as  the  square  of  the  distance.  If  E  is 
the  energy  emitted  from  the  source  per  second,  and  if  r^  and  ra 
are  the  radii  of  the  wave  at  different  distances,  and  7t  and  72  the 
corresponding  intensities. 


Hence  the  intensity  varies  inversely  as  the  square  of  the  distance 
from  the  source,  and  the  amplitude  inversely  as  the  distance. 

References 

FLEMING'S  Waves  and  Ripples  in  Water,  Air,  and  Ether  is  an  excellent 
popular  description  of  all  kinds  of  waves. 


198  WAVE  MOTION 

EDSBR'S  Light,  chapters  on  Wave  Motion. 
DANIBLL'S  Principles  of  Physics,  chapters  on  Wave  Motion. 
WOOD'S  Physical  Optics,  Ch.  3  and  4,  gives  an  interesting  account  of  the 
photography  of  sound  waves. 

V      '         Problems 

1.  A  mass  of  196  grams  is  suspended  by  a  rubber  band  of  such  elasticity 
'  that  an  additional  weight  of  5  grams  will  stretch  it  1  cm.     It  is  extended 

1  cm.  and  released.    Find  the  period,  and  the  dis- 
Simple  Harmonic      placement,    velocity,  and   acceleration   9    seconds 
Motion.  after  it  passer  upward  through  its  resting  point. 

Ans.   Z1- 1.256  sec. 

3 -sin  59°.6 «0.862  upward. 

t>— 2.45  cm./sec.  upward. 

a —21. 55  cm./sec.3  downward. 

2.  Water  or  mercury  in  a  U-tube  is  disturbed.     Show  that  the  liquid 
executes  a  simple  harmonic  motion  of  period  T  =*2n\/l/2g,  where  I  is 
the  length  of  liquid  from  surface  to  surface  around  the  bend. 

3.  Compound  two  simple  harmonic  motions  of  same  period  and  in  same 
plane  with  amplitudes  3  and  2  and  with  phase  difference  of  one-sixth  of 
a  period.  Ans.  72-4.36' 

4.  Compound  two  simple  harmonic  motions  at  right  angles  with  periods 
in  the  ratio  3  :  5  and  with  phase  difference  zero. 

6.  Compound  three  trains  of  waves  of  lengths  in  the 
Waves.  ratios  1,  $,  and  £  and  of  amplitudes  3,  2,  and  1, 

starting  in  the  same  phase. 

6.  Compound  two  trains  of  waves  of  lengths  5  and  4  and  of  equal  amplitudes. 

7.  A  copper  wire  (/o— 8.8)  1  square  mm.  in  cross-section  is  subject  to  a 
tension  of  88,000  dynes.     With  what  velocity  will  a  transverse  wave 
travel  in  it?  Ans.  1000  cm./sec. 

8.  With  what  velocity  will  a  longitudinal  wave  travel  in  the  same  wire? 

Ans.  350,000  cm./sec. 


HEAT 

BY  CHARLES  E.  MENDENHALL,  PH.  D. 
Professor  of  Physics  in  the  University  of  Wisconsin 

INTRODUCTION 

260.  Early  Ideas. — The  preceding  sections  have  dealt  with 
physical  changes  involving,  in  general,  motion  and  changes  in 
motion  of  bodies  as  a  whole.  We  have  now,  however,  to  con- 
sider changes  in  physical  condition  which  do  not  involve  obvious 
changes  in  motion,  of  which  the  most  common  are  changes  in 
hotness  or  coldness  and  changes  in  state,  that  is,  melting  or 
boiling.  The  sense  of  touch  is  the  first  and  simplest  means  of 
distinguishing  hot  from  cold  bodies,  and  by  it  we  can  roughly 
arrange  bodies  in  the  order  of  their  hotness,  deciding  that  A  is 
hotter  than  B,  B  than  C,  etc.  But  the  sense  of  touch  is  found  to 
be  neither  reliable  nor  delicate  enough  to  be  used  as  a  measure 
of  degrees  of  hotness,  and,  moreover,  a  limit  of  hotness  or  coldness 
is  very  soon  reached  beyond  which  the  touch  sense  cannot  be 
directly  applied.  A  purely  physical  basis  of  measurement  ( §264) , 
depending  on  the  properties  of  bodies,  is  therefore  adopted 
which  agrees  with  the  sense  of  hotness  as  far  as  they  can  be 
compared.  When  measured  in  this  definite  physical  way  the 
hotness  of  a  body  is  called  its  temperature,  the  scale  of  measure- 
ment being  so  chosen  that  hotter  bodies  have  higher  temperatures. 

It  is  found  that  increase  in  temperature  of  a  given  body  can  be 
produced  by  various  common  causes,  such  as  contact  with  or  ex- 
posure to  fire,  contact  with  a  hotter  body,  and  friction,  as,  foi 
example,  rubbing  ones  hands  together.  The  causes  which  will 
produce  increase  in  temperature  will  also,  under  proper  condi- 
tions, produce  melting  or  boiling  and  various  other  physical 
changes,  of  which  increase  in  size  is  the  most  common  and  ob- 
vious. On  account  of  these  common  causes  it  was  most  natural  to 
group  together  the  various  effects  referred  to  as  they  became 
known  and  attribute  them  all  to  the  passage,  into  or  out  of  bodies, 

199 


200  HEAT 

i 

of  a  substance  called  caloric  or  heat,  the  presence  or  absence  of 
which  accounted  for  all  of  these  related  phenomena.  According 
to  this  theory,  heat  was  a  material  substance,  but  one  which 
could  not  be  weighed  or  detected  by  any  ordinary  physical 
method.  On  the  basis  of  this  hypothesis  fairly  consistent  ex- 
planations were  given  for  many  common  facts.  For  example,  the 
temperature  of  a  body  was  said  to  depend  on  the  amount  of 
caloric  it  contained  and  upon  its  natural  capacity  for  caloric, 
which  in  turn  depended  upon  its  physical  state,  as,  for  instance, 
the  state  of  subdivision.  A  given  amount  of  matter  in  pow- 
dered form  was  thus  supposed  to  have  a  less  capacity  for  caloric 
than  the  same  quantity  in  larger  pieces.  Thus  the  rise  in  tern- , 
perature  produced  by  rubbing  two  bodies  together  was  explained 
as  being  due  to  the  abrasion  of  the  material,  its  capacity  for 
caloric  being  thereby  reduced  and  a  certain  proportion  of  its 
caloric  set  "free,"  and  its  temperature  correspondingly  raised. 
According  to  this  idea,  the  entire  amount  of  caloric  set  free  should, 
under  given  circumstances,  be  proportional  to  the  entire  amount 
of  material  abraded. 

261.  Heat  and  Work. — The  first  serious  question  of  the  truth 
of  the  caloric  theory  was  raised  in  1798  by  Count  Rumford  who, 
in.experiments  carried  out  in  Munich  upon  the  caloric  developed  in 
the  boring  of  cannon,  used  a  blunt  borer  which  cut  very  little 
material,  and  arranged  matters  so  that  the  heat  generated  raised 
the  temperature  of  a  considerable  quantity  of  water  which  was 
made  to  boil  "without  fire."  From  these  experiments  he  con- 
cluded, that  the  amount  of  caloric  developed  was  not  at  all  pro- 
portional to  the  amount  of  abrasion  but  was,  at  least  approxi- 
mately, proportional  to  the  amount  of  mechanical  work  required 
to  do  the  abrading.  In  the  following  year  Sir  Humphrey  Davy 
performed  the  similar  but  more  striking  experiment  of  melting 
ice  by  rubbing  two  blocks  of  it  together,  the  temperature  of  the 
ice  as  a  whole  being  below  freezing,  and  again  it  was  concluded 
that  the  melting  was  due  to  the  transmission  of  motion  to  the 
ice  molecules.  From  this  time  on  the  idea  that  heat  could  be 
produced  from  mechanical  motion  and  vice  versa,  or,  as  it  is  put 
to-day,  that  heat  is  a  form  of  energy,  was  gradually  accepted.  But 
it  was  nearly  50  years  before  the  full  significance  of  this  new  point  of 
view  was  appreciated  and  careful  measurements  were  made  by 


MOLECULAR  THEORY  201 

Joule  and  others  of  the  amount  of  work  equal  to  a  given  amount 
of  heat.  This  idea  that  heat  is  a  form  of  energy,  together  with  the 
ideas  of  the  kinetic  theory  of  gases  (§227),  and  the  conception  of 
the  molecular  structure  of  matter  suggested  by  chemical  and 
radioactive  (§580)  investigations,  unite  to  give  the  present 
molecular  or  kinetic  theory  of  heat. 

262.  Molecular  Theory. — According  to  this  point  of  view 
matter  consists  of  units  or  parts  called  molecules,  which  are 
composed  of  smaller  units  of  the  elements  (oxygen,  hydrogen, 
iron,  etc.),  called  atoms,  these  in  turn  containing  still  smaller 
units,  namely  elementary  charges  of  negative  electricity  called 
electrons  (§159)  and  probably  a  nucleus  or  center  of  positive 
electricity.  Very  little  is  known  as  to  the  structure  of  atoms,  but 
the  electrons  in  the  atoms  undoubtedly  move  about  or  vibrate 
very  considerably,  possibly  somewhat  as  planets  move  about  the 
sun,  while  the  atoms  move  about  inside  the  molecule,  and 
molecules  move  inside  the-  mass  of  matter,  with  great  freedom 
when  the  matter  is  gaseous,  with  less  freedom  when  it  is  liquid 
or  solid  (§§157-161).  It  is  also  possible  under  various  conditions 
to  have  electrons  existing  more  or  less  independent  of  atoms  as 
"free"  electrons  or  negative  electric  charges,  the  atoms  which 
have  lost  electrons  then  having  a  positive  electric  charge,  and 
being  ready  to  capture  any  other  electron  which  happens  to 
come  near  enough;  free  electrons  are  characteristic  especially 
of  metals.  Broadly  speaking,  the  addition  of  heat  energy  to  a 
body  either  increases  the  (kinetic)  energy  of  motion  of  its  mole- 
cules or  increases  their  (potential)  energy  of  position,  as  when 
melting  or  boiling  occurs. 

Considering  this  more  in  detail  we  see  that  all  of  the  possible  motions  of 
molecules,  atoms,  and  electrons  would  involve  kinetic  energy.  Moreover, 
it  is  evident  that  changes  of  position  of  molecules,  atoms,  and  electrons  with 
respect  to  each  other,  against  whatever  forces,  electrical  or  "  chemical,"  may 
exist  between  them,  would  involve  doing  work  against  these  forces,  that  is, 
changes  in  potential  energy.  Hence  we  can  see  that,  when  heat  energy  is 
added  to  a  body,  it  may  appear: 

1.  As  an  increase  in  the  kinetic  energy  of  motion  of  the  molecules  and  free 
electrons. 

2.  As  an  increase  in  the  potential  energy  of  the  molecules  with  respect  to 
each  other,  in  case  the  average  distance  separating  them  is  increased. 

3.  As  an  increase  in  kiuetio  and  potential  energy  of  atoms  and  electrons 
inside  the  molecules. 


202 


HEAT 


This  analysis  of  the  possible  changes  in  what  is  called  the  internal  energy 
of  bodies  should  be  kept  in  mind  throughout  the  study  of  heat,  which  will  be 
found  to  be  largely  a  study  of  the  effects  of  changes  in  internal  energy  upon 
the  condition  and  properties  of  matter. 


THERMOMETRY 

263.  Standard  Scale  of  Temperature. — We  shall  throughout  use 
the  term  temperature  to  mean  a  quantity  which  we  are  now  to 
define  and  which  can  be  measured  for  any  body  at  any  time. 

Differences  of  temperature  are 

to  agree  with  our  ordinary 
ideas  of  differences  of  hotness 
or  coldness,  so  far  as  the  two 
can  be  compared.  The  scale 
of  temperature  which  we  shall 
adopt  is  the  international  legal 
standard  and  is  based  upon 
the  effect  of  increase  in  hot- 
ness  upon  the  pressure  of  hy- 
drogen. Changes  oj  tempera- 
ture are  defined  as  being  pro- 
portioned to  the  corresponding 
changes  of  pressure  in  a  con- 
stant mass  oj  hydrogen  confined 
at  constant  volume.  This  is 
called  the  hydrogen  constant 
volume  scale.  To  measure  the 
temperature  of  a  body,  for  example,  of  a  barrel  of  water,  the 
vessel  containing  the  hydrogen  would  be  held  in  the  water  and 
the  pressure  of  the  hydrogen  measured.  But  before  tempera- 
ture can  be  expressed  as  a  number ,  we  must  have  a  unit  in 
which  to  express  it  and  we  must  also  agree  on  a  reference  point 
or  "zero"  from  which  it  is  to  be  measured.  The  ordinary  zero 
(reference  point)  is  taken  as  the  temperature  of  a  mixture  of 
pure  ice  and  water  when  the  pressure  on  the  water  surface  is 
1  atmosphere,  while  the  degree  is  fixed  by  adopting  a  second 
standard  point,  the  temperature  of  boiling  water  when  the  pres- 
sure is  1  atmosphere,  which  is  defined  as  + 100°  or  100°  above 
zero.  The  degree  is  then  such  a  change  in  temperature  as  will 


Fio.  170. — Constant  volume  gas  thermometer- 


THERMOMETRY  203 

produce  T^  the  change  in  pressure  which  is  observed  when 
the  hydrogen  is  heated  from  the  -freezing-  to  the  boiling-point  of 
water.  These  specifications  define  the  Centigrade  Zero  and  Centi- 
grade degree,  which  are  universally  used  in  scientific  work. 

A  thermometer  is  an  instrument  for  measuring  temperature 
according  to  some  definite  scale.  A  constant  volume  gas  ther- 
mometer is  an  apparatus  for  measuring  temperature  by  the 
variation  in  pressure  of  a  gas  confined  at  constant  or  nearly 
constant  volume.  If  the  gas  used  is  hydrogen  the  thermometei 
gives  at  once  standard  temperature;  with  other  gases  it  must 
be  calibrated  in  terms  of  the  standard.  Such  an  arrangement 
is  shown  diagramatically  in  Fig.  170,  and  consists  essentially 
of  a  bulb  of  glass,  glazed  porcelain,  fused  quartz,  platinum  or 
platinum-iridium  (according  to  the  temperature  range  over  which 
it  is  to  be  used),  connected  by  a  capillary  tube  to  a  mercury 
pressure-gauge  such  as  the  open  manometer  shown.  The  pres- 
sure of  the  confined  gas  can  be  measured  by  reading  the  differ- 
ence in  level  of  the  two  mercury  columns  and  adding  to  this 
the  atmospheric  pressure  as  determined  simultaneously  with  a 
barometer. 

Still  keeping  the  pressure  of  hydrogen  at  constant  volume  as 
the  basis  of  the  temperature  scale,  other  numbers  may  be  assigned 
to  given  temperatures  by  giving  another  number  to  the  melting- 
point  and  subdividing  the  interval  from  melting  to  boiling  into 
a  different  number  of  degrees.  In  this  way  the  Fahrenheit  scale 
(the  one  in  ordinary  use  in  English-speaking  countries)  is  obtained 
by  giving  the  value  32  to  the  freezing-point  and  subdividing 
the  interval  from  the  freezing-point  to  the  boiling-point,  the 
fundamental  interval  as  it  is  called,  into  180°.  (However,  Fahren- 
heit originally  used  other  temperatures  to  define  his  scale,  namely 
a  freezing  mixture  of  water,  ice,  and  salt  giving  what  he  called 
0°,  an4  blood  heat  which  he  called  96°.)  From  the  above  state- 
ments we  derive  the  following  conversion  formula  for  changing 
from  one  temperature  scale  to  the  other: 


It  must  be  clearly  understood  that  the  choice  of  a  thermometric 
property  (in  this  case  pressure  of  hydrogen)  is  entirely  independent 


204 


HEAT 


of  the  choice  of  numerical  scale,  i.e.,  reference  point  and  size  of 
degree;  the  Centigrade  or  Fahrenheit  numerical  scale  can  each 
be  applied  to  any  other  thermometric  property  desired. 

It  is  found  that  the  change  in  pressure  (volume  constant)  of 


hydrogen  for  1°C.  as  above  defined  is  very  closely 


of  the 


pressure  at  0°C.;  hence  if  the  same  scale  of  temperature  were 

carried  below  zero  Centigrade 
(Fig.  171)  the  pressure  would  be 
reduced  to  zero  at  a  temperature 
of  about  -273.0°C.  This  is 
called  the  absolute  zero  of  the 
hydrogen  constant  volume  scale, 
and,  according  to  the  ideas  of 
the  kinetic  theory  of  gases  (§227)  , 
it  corresponds  to  a  state  of  zero 
molecular  velocity,  .since  pres- 
sure is  due  to  the  impact  of  mov- 
ing molecules.  This  tempera- 
ture could  not,  however,  be 
measured  with  the  hydrogen 
thermometer,  because,  as  we 
shall  see,  the  gas  would  become 
liquid  before  this  point  was 
reached.  We  shall  use  T  to 
represent  absolute  temperatures 
on  the  hydrogen  scale.  In  order 
to  give  at  once  some  idea  of 

the  known  range  of  temperatures  on  the  centigrade  hydrogen 

scale  it  may  be  noted  that: 


Fio.  171. — Temperature  scale  deter- 
mined by  change  in  pressure  of  a  gas  at 
constant  volume.  PQ  —1  Atm.  —  external 
pressure. 


-273.0°  =  absolute  zero. 
—  270°=  lowest  temperature  ever  measured. 
— 190°  =  temperature  of  liquid  air  under   1    atmosphere 

pressure. 
—  80°  =  lowest  recorded  natural  temperature. 

0°  =  melting-point  of  ice. 

100°  =  boiling-point  of  water  under  1  atmosphere  pres- 
sure. 


THERMOMETRY  205 

700°  =  "dull  red"  heat  for  most  solids. 
1400°  =  "white  heat"  for  most  solids. 
3800°  =  about  the  temperature  of  the  electric  arc. 
6000°-7000°  =  Sun's  temperature. 

264.  Constant  Volume  Gas  Thermometer. — In  order  to  use  the 
constant  volume  gas  thermometer  in  the  simplest  possible  way  to 
measure  temperatures  according  to  the  standard  hydrogen  scale, 
it  is  evident  that  the  volume  of  the  bulb  should  be  absolutely 
constant,  and  that  all  the  gas  used  (including  that  in  the  capillary 
and  over  the  mercury)  should  be  heated  to  the  temperatures  to 
be  measured.  This  is  impracticable,  and  hence  corrections  must 
be  made  to  the  observed  readings.  Disregarding  all  corrections, 
we  derive  an  approximate  expression  for  the  temperature  of  the 
bulb  corresponding  to  a  given  pressure  reading,  as  follows: 
Let  P0  =  pressure  of  hydrogen  at  the  freezing-point  of  water, 

P100  =  pressure  of  hydrogen  at  the  boiling-point  of  water, 
t  —  some  other  temperature  of  the  bulb,  the  value  of  which 

is  to  be  determined,  measured  from  centigrade  zero. 
Pt    =  pressure  of  hydrogen  at  this  temperature  t, 

Then,  in  accordance  with  the  definition  of  the  degree  (§263),  we 
define  any  temperature  t  on  the  Centigrade  scale  of  the  constant 
volume  hydrogen  thermometer  by  the  following  formula: 

Pt-P0        Pt-P0 

—  T>  T>    —       J»Z> 

•*  100 — -to  OJr  o 

100 

p  00_p 

where  6  stands  for  the  fraction  °>  which,  for  hydrogen,  is 

lUUxo 

.,  increase  in  pressure  for  1°C. 

the  same  as  -  ,  n9f*— 

pressure  at  0  C. 

If  now  t0  is  such  that  Pt  =  0,  it  follows  that, 

Po  _1 

0  —        T>  E>  7»' 

*100  —  f0  0 


"100 

That  is  to  say,  the  absolute  zero,  at  which  P  =  0,  is  1/6  degrees 
Centigrade  below  0°C.     Therefore  letting  T  represent  the  value 


206  HEAT 

of  a  temperature  measured  from  absolute  zero  we  have 


To  being  'the  number  on  the   absolute  scale  corresponding  to 
0°  on  the  Centigrade  scale. 

The  constant  b  is  called  the  "coefficient  of  increase  of  pressure" 
or  simply  the  pressure  coefficient;  for  hydrogen  its  value  is 
2731.oi»  hence  the  value  of  the  absolute  zero  of  temperature  on  the 
centigrade  constant  volume  hydrogen  scale  as  defined  above 
would  be  -273.04°.  The  value  of  6  for  air  and  nitrogen  also  is 
not  very  different  from  -yfy,  so  that  these  two  gases  would  give 
constant  volume  temperature  scales  approximately 
agreeing  with  the  standard.  Nevertheless  it  is 
obvious  that  the  exact  definition  of  the  standard 
scale  as  here  given  is  entirely  dependent  upon  the 
properties  of  hydrogen.  It  has  been  found  impos- 
sible, however,  to  use  hydrogen  above  about  1100° 
C.  because  of  the  ease  with  which  it  passes  through 
the  walls  of  the  metal  bulbs  which  are  best  used  at 
higher  temperatures;  under  these  conditions  nitro- 
gen is  usually  substituted. 

All  gases  increase  in  pressure  if  heated  at  con- 
stant volume,  and  the  pressure  at  any  temperature 
is  given  approximately  by  the  equation 


FIO  172  _     where  P0  is  the  pressure  at  0°C.  and  b  has  some- 
Constant  pres-     what  different  values  for  different  gases  (§279). 

sure  gas  ther- 
mometer. 265.  Constant  Pressure  Gas  Thermometer. — The  constant 
pressure  gas  thermometer,  which  makes  use  of  the  increase 
in  volume  with  increasing  temperature  of  a  gas  confined  at  constant  pres- 
sure, is  a  convenient  indicating  device  for  demonstration  purposes,  though 
seldom  used  for  precise  measurements.  As  shown  in  Fig.  172,  the  constant 
pressure  used  is  that  of  the  external  atmosphere,  and  the  change  in  volume 
is  proportional  to  the  motion  of  an  indicating  globule  of  mercury  or  other 
liquid  along  a  tube  of  uniform  tore. 

The  coefficient  of  expansion,  that  is  to  say,  the  ratio    ^"fr  '*'  where  V9,  V100 
are  the   volumes  at  0°  and   100°C.    respectively    (pressure   constant),  is 
approximately     *H  for  hydrogen,  air,  oxygen  and  nitrogen,  so  that  an . 
extremely  sensitive  indicator  may  easily  be  obtained. 


THERMOMETRY 


207 


With  a  bulb  about  10  cm.  in  diameter  and  a  tube  5  mm.  in  diameter  the 
motion  of  the  globule  would  be  about  10  cm.  per  degree  change  in  tempera- 
ture of  the  bulb.  The  expansion  of  air  when  heated  is  one  of  the  earliest 
known  effects  of  heat,  and  the  first  thermometer,  invented  by  Galileo  in 
1593,  was  based  on  this  principle. 

266.  Mercury  Thermometers. — For  ordinary  purposes  ther- 
mometers depending  on  the  expansion  of  mercury  confined  in  a 
bulb  and  tube  of  glass  or  other  transparent  substance  are  most 
convenient  and  universally  used.  Two  standard 
forms  are  shown  in-  Fig.  173,  the  mercury  being 
confined  in  a  thin-walled  glass  bulb  attached  to  an 
extremely  fine  capillary  tube.  For  use  at  ordinary 
temperatures  the  upper  part  of  the  capillary  con- 
tains only  mercury  vapor.  Since  mercury  expands 
somewhat  less  than  -^^  part  of  its  volume  at  0°C. 
for  a  degree  rise  in  temperature  (compare  with 
air  above),  it  is  necessary  to  have  a  very  fine 
capillary  in  order  to  obtain  an  easily  observable 
motion  of  the  column  for  a  degree  change  in  tem- 
perature. All  such  thermometers  should,  for  pre- 
cise work,  be  calibrated  or  standardized  by  com- 
parison with  the  hydrogen  standard. 

Fig.  173  shows  the  two  standard  ways  of  marking  the 
"scale"  on  the  thermometer.  In  one  the  scale  is  marked 
directly  on  the  stem  of  the  thermometer — this  is  the  most 
accurate  and  permanent  way,  used  in  all  standard  scientific 
thermometers  and  clinical  thermometers;  in  the  other  the 
scale  is  on  paper  or  white  glass  and  enclosed  in  an  outer 
glass  tube  back  of  the  capillary  stem — this  usually  gives 
more  legible  scales  but  they  are  somewhat  likely  to  become 
loose  and  shift  with  respect  to  the  capillary.  A  third 
method  is  used  for  cheap  "household"  thermometers;  in 
this  the  thermometer  is  simply  mounted  on  a  support  which 
carries  the  scale. 

The  glass  used  for  the  thermometer  (especially  the  bulb) 
is  of  the  greatest  importance,  and  in  recent  years  great  im- 
provements have  been  made  in  the  qualities  of  glass  used 
for  this  purpose.     A  bulb  made  of  ordinary  glass  has  the 
fault  of  slowly  changing  its  volume  with  time,  and  of  permanently  and 
quickly  increasing  its  volume  whenever  it  is  heated,  say  to  100°C.  or  higher. 
Such  changes,  of  course,  alter  the  reading  for  a  given  temperature.     Some 
of  these  effects  gradually  disappear  after  the  bulb  has  been  made;  so  that 


Fio.  173.— 
Thermometers 
with  solid  stem 
and  with  en- 
closed scale. 


208 


HEAT 


thermometer  bulbs  should  be  kept  for  some  time,  or  else  artificially  "aged" 
by  heating  and  cooling,  before  being  graduated. 

Through  the  development  of  special  glasses  having  high  melting-points  it 
has  become  possible  to  construct  mercury-in-glass  thermometers  reading  to 
550°C.  or  even  higher.  In  such  high-range  thermometers  the  space  above 
the  mercury  column  must  be  filled  with  a  gas  (usually  carbon  dioxide  or 
nitrogen)  at  a  final  pressure  of  about  19  atmospheres,  in  order  to  keep  the 
mercury  from  boiling.  For  such  thermometers  the  properties  of  the  glass 
are  of  the  greatest  importance,  and  the  glass  known  as  "Jena  59m"  is  the 
best  one  to  use.  Even  with  this  glass  if  the  thermometer  is  kept  at  550°C. 
for  an  hour  or  more  a  permanent  expansion  of  the  bulb  will  result:  this  will 
permanently  lower  the  freezing-point  reading,  but' if  this  change  is  applied 
as  a  correction  (added)  to  subsequent  readings  of  the  thermometer,  fairly 


Fio.  174. — Maximum  and  minimum  thermometers. 

correct  results  can  be  obtained.  Thermometers  of  mercury  in  clear  fused 
quartz  have  also  recently  been  satisfactorily  constructed  for  use  up  to  about 
700°C. 

In  using  thermometers  it  is  well  always  to  avoid  too  sudden 
heating  or  cooling;  and  in  measurements  above  100°  (or  in  all 
cases  where  extreme  accuracy  is  required)  it  must  be  remembered 
that  thermometers  are  usually  graduated  to  read  correctly  when 
bulb  and  stem  are  all  at  the  temperature  to  be  measured.  If  the 
stem  is  cooler  than  the  bulb  the  thermometer  will  read  too  low  and 
this  error  may  amount  to  as  much  as  40°  at  550°G.  In  careful 
work  thermometers  should  always  be  compared  with  a  standard, 
or  standardized  at  known  temperatures  (§271)  or  sent  to  the 
Bureau  of  Standards  for  comparison. 

267.  Special  Forms  of  Thermometers. — Alcohol  and  some  other 
liquids  have  an  advantage  over  mercury  in  their  greater  coeffi- 
cient of  expansion  and  smaller  surface  tension  (giving  more 


THERMOMETRY  209 . 

regular  rise  and  fall  in  the  capillary),  but  they  are  seldom  used 
for  accurate  thermometers.  Since  mercury  freezes  at  -38.8°C., 
thermometers  containing  alcohol  are  often  used  for  temperatures 
below  this.  Pentane  (C6H12)  is  also  used  for  thermometers 
reading  to  -190°C. 

^  Maximum   and  minimum  thermometers  are  ther- 

mometers provided  with  devices  for  recording  the 
maximum  or  minimum  point  reached  by  the  end  of 
the  mercury  column.  The  maximum  thermometer 
is  usually  of  one  of  two  forms.  In  the  first  form, 
a  small  iron  index  is  pushed  ahead  of  the  mer- 
cury column  and  left  when  the  column  contracts, 
the  lower  end  of  the  index  indicating  the  highest 
reading  of  the  mercury  column;  in  the  second  form, 
Fig.  174,  a  contraction  is  made  in  the  bore  of  the 
tube  near  the  bulb  and  at  this  point  the  mercury 
column  breaks,  when  contraction  occurs  after  the 
maximum  point  is  reached,  leaving  the  upper  end  of 
the  column  at  the  maximum  reading.  This  device 
of  a  contracted  bore  is  used  in  clinical  thermometers, 
Fig.  175.  Minimum  thermometers,  Fig.  174,  are  usually 
of  alcohol  in  glass,  and  have  below  the  meniscus 
a  light  index,  of  such  form  that  the  alcohol  can  flow 
past  it,  while  it  will  be  dragged  down  when  the 
descending  meniscus  reaches  it.  If  the  thermometer 
is  kept  nearly  horizontal,  the  index  will  rest  at  the 
lowest  point  reached  by  the  meniscus. 

For  some  purposes  (especially  common  thermostats) 
FIO.  ITS.—    metallic  thermometers  are  used.     They  usually  de- 
ciinicai       pend  upon  the  bending  of  a  duplex  metallic  bar,  Fig. 
ier^r.me       181,  because  of  the  different  amounts  of  expansion 
of  its  component  metals.     They  are  not  saistfactory 
for  accurate  work. 

268.  Resistance  Thermometry. — In  recent  years  an  electrical 
method  of  thermometry  has  come  into  very  general  use.  In 
this  the  thermometric  property  is  the  resistance  offered  by  a 
metallic  wire  to  the  passage  of  an  electric  current,  which 
resistance  changes  with  the  temperature.  It  must  be  re- 
membered that  such  thermometers  like  all  secondary  instru- 

14 


210 


HEAT 


ments,  must  be  calibrated  in  terms  of  the  hydrogen  standard.  On 
account  of  its  permanence,  high  melting-point  and  acid-resist- 
ing qualities,  pure  platinum  wire  has  been  most  extensively  used 
for  this  purpose,  though  for  use  at  ordinary  temperatures  copper 
and  iron  wire  may  be  substituted.  The  usual  form  of  platinum- 
resistance  thermometer  is  shown  in  Fig.  176,  the  coil  whose 
resistance  changes  are  to  be  measured  (called  by  analogy  the 
"bulb"  of  the  thermometer),  being  mounted  in  a  protecting 
tube  of  glass,  or  (which  is  better)  of  metal 
for  moderate  temperatures  and  of  porcelain 
for  high  temperatures. 

The  advantages  of  the  platinum  thermome- 
ter are  permanence  and  reliability,  wide  range 
(it  may  be  used  up  to  1200°C.),  the  fact  that 
the  readings  may  be  made  at  a  distance  of 
several  hundred  feet  from  the  thermometer 


Fio.  176— Plati- 
num resistance 
thermometer. 


Fio.  177. — Wheatstone's  bridge  for  measuring 
resistance  of  platinum  thermometer. 


itself  and  that  it  may  be  made  accurately  self  recording.  It 
is  also  capable  of  extreme  sensitiveness,  -nrJWC.  being  .read- 
able. For  all  these  reasons  its  use  in  scientific  and  engineer- 
ing work  is  rapidly  increasing.  * 

Fig.  177  shows  the  electrical  leads  to  the  coil  Lj  L2,  compensating  Jeads 
L,  L4  by  means  of  which  the  effect  of  temperature  changes  in  the  leads  L, 
L,  are  eliminated,  and  the  connection  of  the  Wheatetone  bridge  (see  §456) 
by  which  the  resistance  is  measured.  From  an  empirical  formula  developed 


THERMOMETRY  211 

by  Callendar  the  temperature  corresponding  to  a  given  resistance  may  easily 
be  obtained.  This  formula  is  of  such  a  form  that  only  three  known  tempera- 
tures are  needed  to  determine  its  constants.  It  is,  therefore,  very  easy  to 
standardize  a  platinum  thermometer. 

269.  Thermo-electric    Thermometer. — When    two     different 
metals  are  joined  together  in  a  circuit  as  shown  in  Fig.  178,  and 
one  junction  is  heated,  an  electro- 
motive force  is  in  general  produced 

(see  §477),  which  tends  to  drive 
a  current  in  a  certain  direction  as 
shown  and  this  electromotive  force 
increases  as  the  difference  in  tem- 
perature between  the  two  junctions 
increases.  This  thermal  electromo- 
tive force  is  another  thermometric  FIO.  178.— Thermoelectric  couple, 
property  very  extensively  used.  For  «bomag^rection  of  current  produced 

some  purposes  a  voltmeter  (see  § 

453)  suffices  to  measure  the  electromotive  force  generated  by 
heating  one  junction,  and  it  may  be  calibrated  to  read  tempera- 
ture directly.  The  thermoelectric  thermometer  or  thermo-couple, 
as  it  is  called,  is  valuable  on  account  of  its  sensibility,  quick 
response  to  temperature  changes,  and  the  small  size  and  mass 
of  the  part  which  must  be  heated  as  compared  to  the  bulb  of  a 
mercury  or  resistance  thermometer. 

For  work  below  500°C.,  wires  of  copper  and  constantan  (an  alloy  of  copper 
and  nickel)  are  quite  satisfactory;  up  to  1000°C.  wires  of  nickel  and  nickel- 
chromium  alloy  may  be  used  for  approximate  work,  while,  for  the  entire 
range  up  to  1600°C ,  the  most  accurate  results  are  given  by  wires  of 
platinum  and  platinum  +10%  rhodium. 

270.  Measurement  of  High  and  Low  Temperatures. — The  measurement 
of  extreme  high  or  low  temperatures  presents  separate  and  difficult  problems. 
This  is  partly  because  of  mechanical  difficulties  caused  by  changes  in  prop- 
erties of  ordinary  substances  at  extreme  temperatures  (for  example,  melt- 
ing and  softening  of  metals  and  porcelain),  and  chemical  reactions  at  high 
temperatures,  and  partly  because  the  range  of  the  direct  hydrogen  thermome- 
ter is  passed  and  it  is  necessary  to  extrapolate  by  means  of  some  empirical 
formula.    At  high  temperatures  recourse  is  had  to  nitrogen  in  a  constant 
volume  thermometer,  which  has  been  used  from  1100°C.  to  1550°C.;  above 
this  for  a  short  range  thermoelectric  extrapolation  is  possible,  while  beyond 
this  a  radiation  scale  (see  §339)  and  radiation  methods  are  the  only  re- 
sources.    At  low  temperatures  the  least  liquefiablc  gas,  helium,  used  in  a 


212  HEAT 

constant  volume  thermometer,  but  at  a  pressure  of  only  10  cm.  of  mercury, 
has  been  used,  as  well  as  the  resistance  and  thermoelectric  methods. 

271.  Standard  Temperatures. — For  the  purpose  of  standardizing  ther- 
mometers, thermo-couples,  resistance  thermometers,  etc.,  it  is  convenient 
to  make  use  of  one  or  more  temperatures  which  can  easily  be  obtained  and 
kept  constant,  and  which  have  been  accurately  measured.    For  such  pur- 
poses melting-  and  boiling-points  are  the  most  convenient.     To  use  a 
standard  boiling-point  the  liquid  must  be  steadily  boiled  at  a  known  pres- 
sure and  the  thermometer  immersed  in  the  vapor;  to  use  a  melting-point  the 
thermometer  may  be  immersed  in  a  mixture  of  the  solid  and  liquid.     The 
following  table  gives  some  of  the  more  useful  points. 

TABLE  1 

STANDARD  TEMPERA-TUBES 
(Pressure  Constant  at  One  Atmosphere) 

Hydrogen  (liquid) Boiling-point,  -253°C. 

Oxygen .' .  Boiling-point,  — 183 

Carbon  dioxide Boiling-point,  —  78 . 2 

Mercury Melting-point,  —  38 . 8 

Water Melting-point,  0 

Ether Boiling-point,  34.6 

Alcohol  (ethyl) Boiling-point,  78 . 3 

Water Boiling-point,  100 

Napthalene Boiling-point,  218.0 

Tin Melting-point,  231.9 

Benzophenone Boiling-point,  306 . 0 

Sulphur Boiling-point,  444. 7 

Sodium  chloride Melting-point,  801 

Silver Melting-point,  960 

Gold Melting-point,  1063 

Palladium Melting-point,  1549 

Platinum Melting-point,  1753 

272.  The    Pressure,  Volume,   Temperature   Diagram. — From 
the  discussion  of  §262  we  saw  that  in  order  to  know  the  condition 
of  a  body  we  should  know  the  amount  of  energy  present,  per  unit 
mass,  in  several  different  forms,  namely  as  kinetic  energy  of 
molecules,  atoms  and  electrons  and  as  potential  energy  of  mole- 
cules, atoms  and  electrons.     Of  the  entire  amount  of  this  internal 
energy  we  have  no  knowledge,  but  we  can  measure  the  heat 
energy  which  passes  into  or  out  of  a  substance  and  also  the 
external  work  done,  which  together  constitute  the  change  in  the 
internal  energy,  and  hence  we  can  tell  when  a  body  is  brought 
back  to  a  given  condition  of  total  internal  energy.     Now  it  is 
found  that  in  the  majority  of  cases  when  a  body  is  brought  back 


THERMOMETRY  213 

to  the  same  total  energy  content  its  pressure,  volume  and  tem- 
perature return  to  the  same  values,  and,  in  fact,  all  its  physical 
properties  are  the  same  as  before;  hence  it  is  said  that  the 
pressure,  volume,  and  temperature  determine  the  physical  state 
of  a  body. 

These  three  variables,  P,  V,  and  t  are,  however,  not  independent  but  are 
connected  by  a  relation,  called  an  equation  of  state,  the  general  form  of  which 
is  not  known.  This  relation  expresses  the  experimental  fact  that  if  we 
fix  any  two  of  the  three  variables,  P,  V,  and  tt  the  third  must  have  a 
definite  value.  For  example,  if  a  gas  occupies  a  given  volume  at  a  given 
pressure  it  must  have  a  certain  temperature. 

Since  the  physical  condition  of  a  body  is  determined  by  the 
values  of  the  three  variables,  P,  7,  and  ty  it  is  very  natural  to 
represent  a  given  condition  by  a  point  having  the  corresponding 
values  of  P,  7,  and  t  as  coordinates  measured  along  three  rec- 
tangular axes,  as  in  Fig.  179,  where  every  point  in  space  repre- 
sents a  definite  physical  condition.  If  p 
we  take  as  the  origin  absolute  zero  values 
of  P,  7,  and  t,  then  negative  values  of  7 
and  t  will  mean  nothing  physically,  T  p 
while  negative  values  of  P  will  mean 
tensions.  Points  in  a  plane  parallel  to 
the  P7  plane  will  correspond  to  physical 
conditions  for  all  of  which  the  tempera-  * 

i       •      M      i  Fio.  179. — The  use  of  P,  V,  t  as 

ture  is  constant,  and,  similarly,  planes  coordinates, 

parallel  to  the  tV  and  Pt  planes  respec- 
tively will  represent  constant  pressure  and  constant  volume  con- 
ditions. Since  it  is  usually  sufficient  to  fix  two  of  the  varia- 
bles P,  7,  and  t,  physical  conditions  are  often  represented  by 
points  in  a  plane,  for  which  purpose  the  P7,  Pt,  or  Vt  plane  may 
be  chosen. 

EXPANSION 

273.  Introduction. — The  important  changes  in  substances  pro- 
duced by  heat  are  changes  in  size,  changes  in  the  arrangement 
of  molecules  with  respect  to  one  another,  and  changes  in  state, 
from  solid  to  liquid  and  gaseous.  The  difference  between  solids, 
liquids  and  gases  has  been  discussed  in  §157.  Solids  in  general 
offer  great  resistance  to  change  of  shape,  and  their  molecules 


214 


HEAT 


tend  to  assume  a  definite  arrangement  in  groups  called  crystalline 
structure,  not  only  in  obviously  crystalline  minerals  such  as 
quartz,  but  in  all  solids.  The  existence  of  such  structure  is 
sometimes  taken  as  a  test  for  the  solid  state,  though  liquids  also 
can  have  crystalline  properties,  and  it  is  difficult  to  draw  a 
sharp  distinction  between  the  two.  From  the  heat  stand- 
point the  important  matters  are  that  the  average  molecule  in  a 
solid  moves  about  much  less  than  in  a  liquid  or  gas,  and  that  the 
potential  energy  of  the  molecules  with  respect  to  each  other  is 
greatest  in  the  gaseous  state;  furthermore  the  potential  energy 
of  a  solid,  liquid  or  gas  changes  with  its  change  of  size,  or  expan- 
sion due  to  heat.  In  discussing  the  expansion  of  solids  it  is 
convenient  to  consider  both  their  change  in  linear  dimensions 
and  their  change  in  volume,  while  for  fluids  the  latter  alone  has  a 
meaning. 


Fio.  180. — Apparatus  for  measuring  coefficient  of  linear  expansion  of  solids. 

274.  Linear  Expansion  of  Solids. — This  is  an  effect  very  easily 
observed  and  very  widely  made  use  of.  Telegraph  wires  which 
sag  in  summer  are  taut  in  winter;  the  tires  of  wagon  and  loco- 
motive wheels  and  jackets  of  large  cannon  are  made  too  small  to 
slip  in  place  and  are  then  put  on  while  expanded  by  heat,  so  that 
when  cool  and  shrunk  they  have  a  firm  grip.  Different  solids 
expand  differently  for  the  same  change  in  temperature.  A 
simple  experimental  arrangement  for  measuring  the  amount  of 
expansion  is  shown  in  Fig.  180  where  A,  a  bar  of  the  material  being 
studied,  is  supported  in  a  bath  so  that  its  temperature  may  be 
varied.  Two  microscopes,  supported  by  a  frame  distinct  from  the 
bath,  are  arranged  so  that  one  or  both  may  be  moved  parallel  to 
the  bar  by  a  fine  micrometer  screw  and  focused  on  two  fine  marks 


THERMOMETRY  215 

made  on  the  bar.  As  the  bar  expands  the  microscopes  are 
moved  so  that  the  cross-hairs  remain  set  on  the  marks,  and  thus 
the  expansion  can  be  read  from  the  graduated  heads  of  the  microm- 
eter screws.  By  substituting  a  standard  meter  for  the  bar,  the 
actual  length  between  the  marks  at  any  desired  temperature, 
say  0°C.,  may  be  determined,  and,  by  adding  to  this  the  observed 
expansions,  the  length  Lt  of  the  bar  at  any  temperature  t  may 
be  obtained.  The  expansion  will  usually  be  found  to  be  approxi- 
mately, though  not  exactly,  proportional  to  the  change  in  tem- 
perature, that  is  to  say,  if  the  values  Lt  are  plotted  as  ordinates 
with  the  corresponding  values  of  t  as  abscissae,  the  result  will  be 
a  curve,  though  the  curvature  is  usually  slight.  In  general  it  is 
found  that  Lt  may  be  very  closely  represented  by  an  expression 
of  this  form, 

Lt  =  L0(l+at  +  bt* +<**  +  •  •  •)  (1) 

where  a,  6,  c  are  constants  and  t  is  the  temperature  on  the  Centi- 
grade scale.  The  number  of  constants  necessary  increases  with 
the  temperature  range  over  which  it  is  attempted  to  work  and 
with  the  accuracy  desired,  and  varies  also  with  different  sub- 
stances. For  small  temperature  differences  a,  usually  called 
"the  coefficient  of  expansion/'  is  sufficient,  and  its  value  is 
evidently 

Lt-L. 


a  — 


Lj. 


Frequently  also  a  mean  coefficient  of  expansion  between  two  tem- 
peratures, tl  and  t2,  is  used,  and  its  value  is  accordingly 


For  moderate  ranges  of  temperature  (e.g.,  0°  to  100°)  a  and 
am  usually  differ  so  little  that  they  need  not  be  distinguished. 

As  may  be  seen  from  the  following  table,  the  coefficients  of  ex- 
pansion are  never  large,  and  very  refined  experimental  methods 
are  necessary  to  determine  them  accurately,  as,  for  instance, 
some  form  of  interferometer  (§722). 


216  HEAT 


TABLE  2 

COEFFICIENTS  OF  LINEAR   EXPANSION 

Substance.  Cm.  per  degree  C.  per  cm. 

Aluminum 25.5XlO~c 

Brass 18.9  " 

Copper 16.7  " 

Glass  (Jena  16m) 7.8  " 

Gold 13.9  " 

Hard  rubber 80. 

Ice 50.7  " 

Invar 0.7  " 

Iron  (cast) 10.2  " 

Iron  (wrought) 11 .9  " 

Lead 27.6  " 

Nickel 12.8  " 

Oak,  ||  grain 4.9  " 

Oak,    J_  grain 54.4  " 

Platinum 8.9  " 

Porcelain  (Berlin) 2.8  " 

Quartz,  ||  axis 7.5  " 

Quartz,  J_  axis 13.7  " 

Quartz,  fused 0.39  " 

Silver.... 18.8  " 

Tin 21 .4  " 

Zinc 26.3  " 

Isotropic  solids,  including  crystals  in  the  cubical  system 
(with  three  equal  axes  of  symmetry),  expand  equally  in  all  direc- 
tions. Other  crystals  have  one  axis  of  symmetry,  with  one 
coefficient  of  expansion  along  this  axis  and  another  one  in  a  plane 
at  right  angles  to  the  axis,  the  coefficient  being  the  same  in  all 
directions  in  this  plane;  while  still  others  have  three  different 
expansions  along  three  axes,  in  some  cases  even  showing  a  con- 
traction along  one  axis.  In  such  cases  of  unequal  expansions 
the  angles  of  a  crystal  change  as  the  crystal  expands. 

276.  Applications  of  Linear  Expansion. — The  expansion  of 
solids,  especially  the  differential  expansion,  is  made  use  of  in 
metallic  thermometers,  thermographs  and  thermostats.  Usu- 
ally a  compound  strip  of  brass  and  iron,  riveted  together,  is 
fixed  at  one  end  and  arranged  so  that  the  bending  of  the  strip, 
due  to  the  unequal  expansion  of  brass  and  iron,  operates  a  record- 
ing or  indicating  pointer,  or,  in  the  thermostat,  makes  electrical 


THERMOMETRY 


217 


contact  to  right  or  left  and  thus  controls  some  heating  system 
Fig.  181  shows  a  common  form. 

The  balance  wheel  of  watches  has  a  rim 
made  of  a  compound  metal  strip,  as  above 
described,  and  so  arranged  that  a  change  in 
temperature,  by  altering  the  curvature  of 
these  strips,  will  move  a  considerable  part  of 
the  mass  of  the  wheel  to  or  from  the  center, 
thus  altering  the  moment  of  inertia  of  the 
wheel  and  hence  its  period.  In  this  way 
other  temperature  effects  on  the  rate  of  the 
watch,  such  as  change  in  elasticity  of  the 
springs,  change  in  diameter  of  the  balance 
wheel,  and  change  in  viscosity  of  the  oil  in  the 
bearings,  may  be  compensated. 

In  the  mercury  clock  pendulum  shown  in 
Fig.  182  the  length  of  the  reservoir  of  mercury 
is  so  chosen  that  the  expansion  of  the  mercury, 
which  raises  the  center  of  gravity,  just  com- 
pensates for  the  expansion  of  the  supporting 
rod  which  lowers  the  center  of  ^  gravity,  so 

that    the    time    Of    vibration   Will 

not  be  altered  by  changes  in  tern- 
perature.  This  compensation  can  thermostat. 
now  be  accomplished  even  more 
accurately  by  the  use  of  a  specially  worked  nickel- 
steel  alloy,  called  "invar,"  which  has  a  coefficient 
of  only  .00000075  to  .00000015,  or  ^  that  of  brass. 
This  alloy  is  also  valuable  for  making  standard 
meter-bars,  tapes  and  scales  with  lengths  practically 
independent  of  small  temperature  variations. 

The  cracking  of  objects  by  heating,  particularly 
sudden  heating,  is  due  to  unequal  expansion  pro- 
duced by  differences  of  temperature  in  different 
parts.     Porcelain  is  less  liable  to  crack  than  glass 
because  of  its  smaller  coefficient  of  expansion,  and 
thin  glass  than  thick  because  of  the  more  rapid 
equalization  of  temperature.     In  fusing  metals  into  glass  to  make 
an  air-tight  joint,  as,  for  example,  the  leads  into  an  incandescent 


Fio.  182.— 
Mercury  com- 
pensation 
pendulum. 


218  HEAT 

lamp  bulb,  it  is  necessary  to  use  a  metal  having  nearly  the  same 
coefficient  of  expansion  as  glass,  otherwise  cracking  (or  leaking) 
would  occur  when  the  joint  cooled.  As  may  be  seen  from  the 
table,  platinum  is  the  best  metal  for  this  purpose.  There  is  a 
very  striking  difference  between  the  coefficients  of  crystalline  and 
fused  quartz;  the  former  cracks  with  the  slightest  heating,  the 
latter,  because  of  its  small  coefficient,  may  be  taken  from  an 
oxy  hydrogen  flame  and  at  once  plunged  into  liquid  air  without 
cracking. 

276.  Cubical  Expansion  of  Solids.  —  If  Vt  represents  the  volume 
of  a  solid  at  t°G.,  V0  its  volume  at  0°C.;  then  it  is  found  that  in 
general  solids  expand  in  such  a  way  that  V  may  be  represented  as 
a  function  of  t  by  an  equation  similar  to  the  one  used  for  linear 
expansion: 

Vt=Vo(l+a't  +  W  +  crt?)  (I) 

and,  as  before,  the  constants,  &',  c',etc.,  are  much  smaller  than  a', 
so  that  for  small  temperature  changes, 

Vt=V0(l+a't)  (2) 

If  we  now  consider  a  cube  of  the  material  of  length  Lt  on  an  edge, 
we  have, 


approximately,  from  equation  (1),  §  274,  or 


neglecting  higher  powers  of  a;  and,  since  V0  =  L0*t 
3a  =  a',  by  comparison  with  equation  (2) 

That  is  to  say,  the  cubical  coefficient  of  expansion  is  three  times  the 
linear  coefficient,  and  can  be  obtained  from  the  preceding  table. 

277.  Expansion  of  Liquids.  —  The  change  of  volume  of  liquids 
with  temperature  has  already  been  mentioned  as  the  basis  of 
liquid-in-glass  thermometers.  The  fact  that  the  mercury  or 
alcohol  in  such  thermometers  rises  with  increased  temperature 
shows  that  the  liquids  expand  more  than  the  glass,  and  this  is 
usually  true  of  liquids  as  compared  with  solids.  To  represent  the 
volume  Vt  of  a  liquid  at  a  temperature  t  in  terms  of  the  volume 
V0  at  0°C.,  it  is  found  that  an  equation  of  the  same  form  will 
suffice  — 


THERMOMETRY 


219 


or  approximately, 


since  b"  is  usually  much  smaller  than  a". 

A  bulb  with  a  capillary  stem  like  a  thermometer  is  usually  used 
in  measuring  the  differential  expansion  of  a  liquid  and  a  solid. 
The  walls  of  the  bulb  expand  as  if  they  we're  filled  with  solid 
material;  hence  the  volume  of  the  bulb  space  is  at  any  tempera- 
ture equal  to  the  expanded  volume  of  the  solid  which  would  fill 
it  at  0°C.  If 

V0  —  volume  of  bulb  and  of  liquid  filling  bulb  at  0°C., 
V't  =  volume  of  bulb  at  t°C., 
Vt  =  volume  of  same  liquid  at  t°C., 

a',  o"  =  volume  coefficients  of  expansion  of  solid  composing  the 
bulb,  and  of  the  liquid  respectively,  then 

-f  a"    -  (1  +  a'0]  =  yo(a"-a'*. 


This  differential  or  apparent  expansion ,  Vt—  V1 't,  can  be  measured 
by  noting  the  rise  of  the  liquid  in  the  capillary  stem.  If,  in  addi- 
tion, the  volume  coefficient  a'  of  the  solid 
is  known,  we  can  determine  the  coeffi- 
cient a"  of  the  liquid;  for — 


Fia.  183. — Method  of  mea- 
suring the  absolute  coefficient 
of  expansion  of  mercury. 


The  difference  a"  —  a'  is  called  the  appar- 
ent coefficient  of  expansion  of  the  liquid. 

It  is  possible  to  determine  the  absolute 
coefficient  of  expansion  of  a  liquid,  inde- 
pendent of  the  expansion  of  a  containing 
vessel,  by  a  method  due  to  Dulong  and 
Petit  and  illustrated  in  its  simplest  form 
in  Fig.  183.  Two  vertical  tubes  filled 

with  the  liquid  in  question  are  connected  at  their  lower  extremi- 
ties by  an  accurately  horizontal  tube.  The  vertical  tubes  are  in 
baths  of  some  sort,  so  that  one  can  be  maintained  at  a  tempera- 
ture of  0°G.  and  the  other  at  f°G.  At  the  bases  of  the  two 
tubes  the  pressures  must  be  equal,  otherwise  there  would  be  a 
flow  from  one  to  the  other  through  the  connecting  tube.  The 
pressure  at  the  two  upper  free  surfaces  must  be  the  same,  since 
it  is  that  of  the  external  atmosphere;  hence  the  difference  in 


220  HEAT 

pressure  from  top  to  bottom  of  the  two  columns  must  be  the 
same.     Hence  by  §185, 


and  *-- 

pt     h0 

But  if  F0*=  volume  of  unit  mass  of  fluid  at  0°C.=  — 

Po 

and  7f«=  volume  of  unit  mass  of  fluid  at  t°C.=— 

then.  Vt=V0(l+a"t) 

and  £__i  +  fl*<    *i 

pt 

Hence  '  a' 

This  method  has  been  especially  used  to  determine  the  absolute 
coefficient  of  expansion  of  mercury;  this  being  known,  mercury 
can  be  used  to  determine  the  coefficient  of  expansion  of  solids 
by  the  differential  method.  The  coefficients  of  expansion  of 
liquids  (except  water)  decrease  with  increase  of  the  pressure  at 
which  they  are  observed. 

TABLES 

COEFFICIENTS  OF  CUBICAL  EXPANSIONS  OF  LIQUIDS 
Substance.  Cm».  per  degree  C.  per  cms. 

Alcohol  (ethyl)  ..............................   110.      X  10~6 

Alcohol  (methyl)  .......................  ,  ____   118.          " 

Benzine  ....................................   124.          " 

Mercury  ....................................     18.  18     " 

Paraffin  oil  .....................  .  ............     90.          " 

Pentane  ....................................   159.          " 

Toluene  .....  .  ..............................   109.          " 

Water,  15-100°  ..............................     37.2 

Xylol  ...........  ...........................   101. 

278.  Expansion  of  Water.  —  Water  is  unique  among  liquids  in 
that  it  has  a  maximum  density  at  about  4°C.,  under  1  atmosphere 
pressure,  i.e.,  below  4°C.  it  contracts  with  rise  of  temperature, 
above  4°C.  it  expands. 

This  property,  which  has  very  important  consequences,  is 
clearly  shown  by  Hope's  apparatus,  Fig.  184.  If  the  tank  around 
the  middle  of  the  glass  vessel  be  filled  with  a  freezing  mixture  of 
ice  and  salt,  and  the  vessel  be  filled  with  water  at  a  temperature 


THERMOMETRY 


221 


higher  than  4°  C.,  the  water  in  the  middle  when  cooled  will  become 
denser  and  fall  to  the  bottom,  whereas  the  water  above  the  middle 
will  not  be  disturbed.  Thus 
the  upper  thermometer  will 
indicate  a  practically  station- 
ary temperature  and  the  lower 
one  a  falling  temperature  until 
all  the  lower  half  of  the  vessel 
is  filled  with  water  at  4°C., 
after  which  the  upper  one  be- 
gins to  fall  in  temperature 
until  0°  is  reached  and  freez- 
ing begins  at  the  top,  the 
lower  thermometer  still  indi- 
cating 4°C.  The  water  at 
4°C.  is  most  dense  and  there- 
fore Collects  at  the  bottom  of  Fia  -184— Hope's  apparatus  for  determin- 
•  i  i  ing  the  temperature  of  maximum  density  of 

the  vessel.  water. 

A  somewhat  similar  opera- 
tion goes  on  in  winter  in  ponds  and  rivers  which  are  not  too 
much  disturbed  by  winds  or  currents,  the  densest  water,  at  4°C., 

collects  at  the  bottom,  while 
the  coldest,  at  0°C.,  being 
lighter,  stays  on  top.  Hence 
freezing  occurs  at  the  top, 
unless  the  entire  mass  of 
water  is  cooled  by  currents 
to  near  0°C.,  in  which  case 
freezing  may  occur  on  the 
bottom  or  on  submerged 
solids,  cooled  by  radiation, 
thus  forming  "ground  ice'1 
which  is  of  serious  conse- 
quence in  northern  rivers. 
The  volume  of  1  gram  of 
water  at  various  temperatures  under  1  atmosphere  pressure  is 
given  in  Fig.  185. 

According  to  Amagat  the  temperature  of  maximum  density  falls  frith 
increase  of  pressure,  being  about  2°C.  under  a  pressure  of  93  atmospheres.    li 


MB 

1.04 
1.03 
1.02 
1.01 
1  00 

\ 

\ 

i 

~>ea 
t  V 

fie 

Vo 

fum 

e  o 

f  v 

father 

/ 

a 

inc. 

us 

Tei 

npt 

'mi 

un 

8 

/ 

/ 

/ 

/ 

/ 

/ 

s^> 

^ 

/ 

-10    0    10    20   30   40    50    60   70   80    90  la 

TEMPERATURES 

Fio.  185.  —  Expansion  curve  for  water. 

222 


HEAT 


kept  by  pressure  in  the  liquid  state  water  continues  to  contract  below  0°C. 
and  [to  expand  at  an  increasing  rate  above  100°C.  The  solution  of  various 
salts  in  water  also  lowers  the  temperature  of  maximum  density,  4  per  cent, 
of  dissolved  common  salt  lowering  it  to  —  5.63°C.  The  peculiar  behavior  of 
water  as  regards  its  thermal  expansion  is  due,  according  to  Tamman,  to  the 
existence  at  low  temperatures  of  several  different  kinds  of  water  molecules 
or  groups  of  molecules  which  gradually  break  up  into  one  simpler  kind  as 
the  temperature  is  raised. 

279.  Expansion  of  Gases. — Since  the  effect  of  pressure  on  the 
volume  of  a  gas  is  very  great  (§221),  it  is  evident  that  in  discuss- 
ing the  expansion  of  gases  with  in- 
crease in  temperature  we  must  be 
careful  to  specify  the  pressure  con- 
ditions which  are  to  hold  during 
the  expansion.  The  simplest  con- 
dition is  to  maintain  the  pressure 
constant  and  measure  the  change 
in  volume  of  the  gas  in  a  bulb  by 
allowing  it  to  expand  and  push  out 
a  mercury  piston  in  an  attached 
tube,  as  illustrated  in  Fig.  172.  For 
accurate  work  an  arrangement  such 
as  is  shown  in  Fig.  186  is  necessary, 
and  for  more  complete  knowledge 
of  the  subject  the  expansion  must 
be  carried  out  at  various  constant 
pressures.  A  correction  must,  of 


"f 

Fio.  186. — Apparatus  for  measuring 
the  expansion  of  gases.  The  gas  ex- 
panding from  the  bulb  in  steam  is 
measured  in  the  graduated  bulb. 


course,  be  made  for  the  expansion  of  the  bulb  and  for  the  fact  that 
an  increasing  amount  of  the  gas  will  be  in  the  stem  and  hence  will 
not  be  heated.  Gay-Lussac  (1802)  and  Charles  (1787)  inde- 
pendently carried  out  such  experiments,  and  arrived  at  the  "  Law 
of  Charles  and  Gay-Lussac,"  according  to  which  all  the  common 
gases  expand  by  a  constant  fraction  of  their  volume  at  0°/0r  each 
rise  of  1°  in  temperature.  This  fraction  is  about  .003660  (Tfy), 
or  about  the  same  as  the  "pressure  coefficient"  of  a  gas  (§264). 
In  the  form  of  an  equation  this  law  is— 


V«-V.(l+oO 


(p  constant), 


but  it  is  now  known  that  the  law  is  only  approximately  true  and 
that  a  is  not  the  same  for  all  gases.     Furthermore,  a  varies  with 


THERMOMETRY  223 

the  pressure  and  with  the  temperature,  and  is  not,  in  general, 
quite  equal  to  the  pressure  coefficient,  b. 

Later  work  of  Regnault  and  others  has  shown  that  the  coefficients  of 
expansion  of  all  gases  except  hydrogen  increase,  at  ordinary  temperatures, 
with  increasing  density  of  the  gas,  and  that  the  coefficients  for  the  several 
gases  are  more  nearly  alike  and  more  nearly  equal  to  their  "  pressure  coeffi- 
cients" when  the  gases  are  at  low  pressures  or  high  temperatures. 

TABLE  4 
EXPANSION  COEFFICIENTS  AND  PRESSURE  COEFFICIENTS 

Gas.  a.  b. 

Air  0.003671  0.003674 

Carbon  dioxide  0  .  003728  0  .  003712 

Hydrogen  0.003661  0.003662 

Nitrogen  0.003673  0.003672 

Temperature  0°C.-100°C.,  Pressure,  1  atmosphere. 

280.  The  Gas  Equation:  A  Perfect  Gas.—  We  have  seen  (§221) 
that  gases  follow  Boyle's  law  more  or  less  closely,  the  product  of 
the  pressure  and  volume  at  constant  temperature  being  nearly 
constant.  In  §264  we  considered  the  change  in  pressure  with 
temperature  of  a  gas  confined  at  constant  volume,  which  is  given 
approximately  by  the  equation  Pt  =  P0(l  +  bt).  In  §279  we  have 
just  discussed  the  expansion  of  gases,  which  occurs  approximately 
according  to  the  relation  Vt=V0(l  +  at),  the  pressure  being  con- 
stant. It  is  convenient  to  combine  these  statements  into  a 
single  equation,  which  will  then  represent  all  the  relations  which 
approximately  hold  between  the  pressure,  volume  and  tempera- 
ture of  a  gas.  This  may  be  done  as  follows  : 

Let  P0V0  be  the  pressure  and  volume  of  a  given  mass  of  gas  at 
0°C.,  and  let  it  be  heated  at  constant  volume  to  t°C.  Then  we 
have  (§264) 

P,  = 
and  hence 


Again,  starting  at  P0700°,  let  it  be  heated  at  constant  pressure  to 
the  same  final  temperature  t°G.;  then  by  the  law  of  Charles 
(§279) 

7t=70 
and  henoe 


224  HEAT 

Now  let  the  pressure,  P,  and  the  volume,  V,  be  changed  in  any 
way,  the  temperature  remaining  t°C.     Then  from  Boyle's  law 


Hence 

and 

a  =  b 

Also  since  (§264)  t  +  \=T  kt  *  \     -.    (X 

We  may  write  P7  =  P0VJbT 

or  P7  =  J?77  (5) 

It  is  frequently  convenient  to  consider  an  imaginary  ideal  or  per- 
fect gas,  which  exactly  obeys  these  gas  laws,  and  which  also  has 
certain  other  properties  which  will  be  referred  to  later.  The  vol- 
ume and  pressure  coefficients  of  such  a  gas  we  shall  designate  by 
a'  and  &',  absolute  temperature  according  to  this  perfect  gas 
scale  by  T't  and  the  constant  factor  by  R'.  The  gas  law  or  equa- 
tion of  state  for  a  perfect  gas  then  becomes  PV  '  —  R'T'. 

281.  Real  Gases.  —  As  we  have  just  seen,  real  gases  follow  more 
or  less  closely  the  law 

PV  =  RT 

where  T  is  the  temperature  measured  with  a  constant  volume 
hydrogen  thermometer  from  the  absolute  zero  of  the  hydrogen 
scale.  The  approximation  to  the  law  PV  =  RT  is  found  to  be 
very  much  closer  for  high  temperatures  and  low  pressures. 

The  value  of  R  is  different  for  different  gases,  and,  of  course, 
also  for  different  masses  of  any  one  gas.  It  is  customary  to  con- 
sider the  equation  as  applying  to  1  gram  of  gas,  and  R  is  then 
called  the  gas  constant  for  this  gas,  and  is  evidently  equal  to 

of  the  product  P0V0  at  0°C.    For  any  other  mass  of  M 


.  U 

grams  the  constant  in  the  equation  PV  —  RT  will  be  MR,  since 
volumes  are  proportional  to  masses  under  given  conditions. 

It  is  easy  to  see,  in  a  general  way,  why  the  properties  of  real  gases  should 
approach  those  of  a  perfect  gas  at  high  temperatures  and  low  pressures.  For, 
according  to  the  simple  kinetic  theory  (§227),  a  gas  having  no  molecular 
forces,  i.e.,  no  molecular  potential  energy,  and  negligible  molecular  volume,  is 
perfect  in  so  far  that  it  obeys  the  law  PV  —  RT.  Now,  it  is  evident  that  the 


THERMOMETRY 


225 


higher  the  temperature  of  a  real  gas,  the  less  will  be  the  proportion  of  the 
potential  to  the  kinetic  energy,  and  also  that  the  larger  the  volume  of  a  gas, 
other  things  being  equal,  the  less  will  be  the  actual  molecular  volume  com- 
pared to  the  total  volume.  Hence,  as  the  temperature  is  raised,  or  the 
density  diminished,  the  conditions  become  more  nearly  those  assumed  in 
the  simple  kinetic  theory. 

By  making  a  still  further  assumption  equation  (5)  may  be  further  gener- 
alized. According  to  Avogadro's  hypothesis  equal  volumes  of  different 
gases  at  the  same  temperature  and  pressure  contain  equal  numbers  of  mole- 
cules, that  is,  the  total  masses  of  equal  volumes  will  be  proportional  to  the 
molecular  weights  of  the  gases,  or 


where  m^  m2,  mt  are  molecular  weights.  Hence,  if  we  take  m^  grams, 
ra2  grams,  and  ms  grams  (called  gram-molecular-weights)  of  these  gases,  they 
will  occupy  the  same  volume  at  the  same  pressure  and  temperature. 

Hence,  PV  = 

Hence,  m1 

and  R"  is  a  constant  for  all  gases,  whose  value  can  be  at  once  computed. 
For  example,  for  nitrogen  m«=28;  specific  volume  F  =  796.2  c.c.  when  T  = 
273°;  and  P  =  l  atm.  =  1,012,630  dynes/cm.* 
Hence 


R' 


PVm 


=  8.305X10* 


ergs 
degree 


282.  Isothermal  Curves. 
— The  significance  of  the 
equation  PV=RT  can  be 
seen  more  readily  by  graph- 
ical representation  accord- 
ing to  the  method  of  §272. 
Giving  T  some  constant 
value,  Tlt  it  is  evident  that 
Boyle's  law,  PV  —  const., 
is  represented  by  a  rectan- 
gular hyperbola  in  a  plane 
parallel  to  the  PV  plane 
and  cutting  the  T  axis  in 
the  point  7\.  If  a  series  of 
such  hyperbolae  are  located 
for  different  temperatures  and  then  projected  upon  the  PV 
plane  by  dropping  perpendiculars  from  every  point  to  this 
plane,  the  result  is  a  family  of  hyperbolae  each  of  which  can  be 

15 


4  8  12 

VOLUME  IN  LITERS 

Fio.  187. — Isothermal  curves  for  air. 


16 


226  ,        HEAT 

distinguished  by  labeling  it  with  the  temperature  belonging  to  it, 
as  shown  for  air  in  Fig.  187.  Any  curve  showing  the  relation 
between  the  pressure  and  volume  of  a  substance  under  the  con- 
dition T  =  const,  is  called  an  isothermal  curve.  We  accordingly 
conclude  that  isothermal  curves  for  a  perfect  gas  are  rec- 
tangular hyperbolse,  and  that  isothermal  curves  for  real  gases 
approximate  to  rectangular  hyperbolae,  the  approximation  being 
closer  at  high  temperatures. 

283.  Molecular  Energy  and  Temperature. — As  we  have  seen 
(§221)  ordinary  gases  very  approximately  obey  Boyle's  law, 
P7  =  constant  for  constant  temperatures,  and  PV  increases  as 
the  temperature  increases.  Also,  according  to  the  kinetic  theory 

of  gases  (§227),  for  a  simple  ideal  gas  PV  =»-Q—  which  will  be 

o 

constant  if  the  average  random  undirected  kinetic  energy  per 
molecule  is  constant,  and  will  increase  in  proportion  to  the 
average  molecular  energy  JAfv3.  From  these  two  statements 
for  a  real  and  an  ideal  gas  it  is  natural  to  conclude  that  the 
temperature  of  a  real  gas  is,  at  least  approximately,  proportional 
to  the  kinetic  energy  of  molecular  motion,  and  even  to  extend  this 
analogy  to  liquids  and  solids  where  it  has  not  the  same  justi- 
fication. While  the  proportionality  of  mean  molecular  kinetic 
energy  to  temperature  turns  out  to  be  very  closely  true  for  gases, 
and  is  a  very  useful  and  instructive  hypothesis,  nevertheless  the 
complicated  structure  of  real  molecules  (as  compared  with  those 
of  the  ideal  gas)  shows  us  that  the  hypothesis  must  not  be 
taken  too  literally. 

CALOB1METRY 

284.  Unit  of  Heat. — Calorimetry  .is  the  process  of  measuring 
quantities  of  heat.  Obviously  the  first  thing  to  be  decided  upon 
is  the  unit  in  terms  of  which  to  measure,  and  though,  as  has 
been  said,  energy  units  may  be  used,  it  is  often  more  conven- 
ient to  use  a  unit  defined  in  terms  of  heat  phenomena  only. 

In  looking  for  a  purely  thermal  unit  of  heat  it  is  natural 
to  pick  out  some  effect  which  heat  produces,  and  agree 
that  the  heat  unit  shall  be  such  an  amount  of  heat  as  will 
produce  a  specified  amount  of  this  effect  in  unit  mass  of  a 
standard  substance.  The  specified  effect  agreed  upon  is  a  change 


CALORIMETRY  227 

in  temperature  of  1°G.,  and  the  standard  substance  is  water.  To 
be  exact  the  particular  degree  must  be  specified;  hence  we  shall 
define  the  unit  of  heat  as  that  quantity  of  heat  which  will  raise  the 
temperature  of  1  gram  of  water  from  14J  to  15J°C.  This  is  called 
the  Calorie,  or  Cal15. 

The  relation  of  this  thermal  unit  to  the  unit  of  mechanical 
energy  has  been  found  by  experiments  which  will  be  described 
later  (§340).  These  show  that  if  the  "mechanical  equivalent 
of  heat,"  that  is,  the  number  of  work  units  equivalent  to  one 
heat  unit,  be  denoted  by  the  letter  J, 


J  =  4.187  X  107     r- 
calone 

Sometimes  a  mean  calorie  is  also  specified.  This  is  one  one-hundredth  of 
the  heat  required  to  change  1  gram  of  water  from  0°  to  100°C.  It  is  about 
equal  to  1  cal.  Sometimes  the  "large  calorie"  equal  to  1000  calories,  is 
used  as  a  unit,  and  in  engineering  practice  (in  English-speaking  countries) 
the  "  British  thermal  unit"  (B.  T.  U.)  is  employed  and  is  equal  to  the  heat 
required  to  raise  the  temperature  of  1  Ib.  of  water  1°  Fahrenheit.  From  the 
relation  of  the  pound  to  the  gram  and  the  Fahrenheit  to  the  Centigrade 
degree,  it  follows  that: 

1  B.  T.  U.-252  Cal. 

In  British  Thermal  units  and  foot  pounds  J  is  778  ft.-lbs./B.T.U. 

The  most  common  method  of  measuring  quantities  of  heat  in 
calories  is  by  the  "  method  of  mixtures,"  which  consists  in  trans- 
ferring the  quantity  of  heat  to  be  measured  to  a  known  mass  of 
water  and  observing  the  resulting  rise  of  temperature  of  the 
water.  The  heat  may  be  transferred  to  the  water  in  many  ways 
—for  example,  by  dropping  a  piece  of  hot  copper  into  the  water, 
by  pouring  some  hot  liquid  into  it,  or  by  passing  steam  into  it. 
It  is  of  course  simplest  to  use  the  water  at  about  the  temperature 
for  which  the  calorie  is  defined^  as  in  that  case  the  number  of 
grams  of  water  used  multiplied  by  the  number  of  degrees  rise  in 
temperature  will  give  at  once,  to  a  first  approximation,  the 
number  of  calories  which  have  been  added. 

285.  Specific  Heat.  —  If  two  different  masses  of  water  are  ex- 
posed for  the  same  length  of  time  in  just  the  same  way  to  a  steady 
source  of  heat,  it  will  be  found  that  the  temperatures  of  the  two 
will  have  risen  inversely  in  proportion  to  their  masses.  If  the 
same  masses  of  copper  be  treated  in  the  same  way  it  will  be  found 
that  the  rise  in  temperature  will  be  more  than  ten  times  as  great, 


228 


HEAT 


but  again  inversely  proportional  to  the  masses.  From  this  we 
conclude  that  the  temperature  effect  of  a  given  heat  agent  acting 
on  a  body  for  a  given  time  depends  on  the  mass  of  the  body  and 
on  a  factor  which  differs  for  different  substances,  and  which  is 
called  the  specific  heat.  The  specific  heat  of  a  substance  is  defined 
as  the  number  of  calories  required  to  raise  the  temperature  of  1  gram 
of  the  substance  1°C.  The  symbol  for  specific  heat  is  s.  To  be 
exact,  the  particular  degree  must  be  specified  because  the  specific 
heat  varies  with  the  temperature,  for  example  the  number  of 
calories  required  to  raise  1  gram  of  a  substance  from  0°  to  1°  is 
different  from  the  number  required  to  raise  it  from  49°  to  50°. 
For  most  purposes,  however,  and  for  not  too  large  temperature 
differences,  say  from  0  to  100°,  it  is  not  necessary  to  consider 
the  variation  in  specific  heat,  and  it  is  customary  to  speak  of 
the  specific  heat,  meaning  the  mean  value  within  the  range 
considered. 

The  heat  capacity,  S,  of  a  body  (of  any  mass  and  variety  of 
parts)  is  the  number  of  calories  required  to  raise  its  temperature 
1°G.  at  the  mean  temperature  t.  This  will  evidently  depend  on 
the  masses  and  specific  heats  of  the  various  parts  of  the  body,  and 
if  mi,  m2,  ws  and  sx,  s2,  ss,  stand  for  the  masses  and  correspond- 
ing specific  heats  of  the  parts,  we  have 

•  etc. 


1.015 
$  1.010 

i 

1  1.005 

Spec 

if  ic  t 

ieat 

of  M 

ater 

V 

.s 

^ 

*~~ 

\ 

X 

^ 

/ 

-QQS 

"*»», 

—  —  —  — 

^**^ 

10       20 


80      40       50 

TEMPERATURES 


60       70       80 


Fio.  188. — Variation  of  specific  heat  of  water 
with  temperature. 


The  Variation  of 
the  Specific  Heat  of 
Water. — The  common 
occurrence  of  water  and 
its  physical  and  chemical 
characteristics  make  it 
extremely  useful  in  heat 
measurements,  hence  a 
knowledge  of  its  specific 
heat  at  various  tempera- 


tures is  of  importance.  The  specific  heat  of  water  is,  of  course, 
unity  at  the  temperature  for  which  the  calorie  is  defined  (§284). 
At  other  temperatures  it  may  be  either  greater  or  less  than 
unity.  The  first  satisfactory  study  of  the  variation  of  the 
specific  heat  was  that  of  Rowland  in  1878;  combined  with  later 


CALORIMETRY 


229 


work,  it  shows  that  the  specific  heat  diminishes  with  rising  tem- 
perature, reaching  a  minimum  between  25°  and  30°C.,  as  shown 
in  Fig.  187.  The  mean  value  of  the  specific  heat  from  0°  to 
100°C.  differs  very  little  from  1. 

287.  Method  of  Mixtures.  —  Returning  now  to  a  more  detailed 
consideration  of  the  method  of  mixtures,  there  are  in  practice 
several  additional  points  to  be  considered,  as  can  best  be  seen 
by  discussing  a  particular  form,  of  apparatus  shown  in  Fig.  189. 
In  the  first  place  the  water  must  be  held  in  some  vessel  G,  con- 
taining a  stirrer  and  a  thermometer,  and  called  a  calorimeter. 
Into  this  some  of  the  heat  will  pass, 
raising  its  temperature.  Moreover, 
some  heat  will  pass  out  of  the 
water  and  containing  vessel  during 
the  operation  and  will,  therefore, 
fail  to  produce  its  proportionate 
temperature  change.  To  take  ac- 
count of  the  first  effect  we  must 
know  the  heat  capacity  of  the 
calorimeter.  The  second  effect, 
loss  of  heat  to  the  surroundings, 
necessitates  what  is  called  the  cool- 
ing or  radiation  correction.  Neg- 
lecting this  correction  for  the  mo- 
ment we  can  write  the  fundamental 
equation  of  the  mixture  calorimeter 
thus 


Stirrer, 


Fio.  189. — Calorimeter  for  method 
of  mixtures. 


which  expresses  the  fact  that  the  heat  added  (H)  equals  the  heat 
gained  by  the  calorimeter  and  water.  In  this  expression  s',  st  and 
s2,  etc.,  are  respectively,  the  mean  specific  heats  of  water  and 
of  the  various  materials  of  the  calorimeter,  m  the  mass  of 
the  water,  ^slml  the  heat  capacity  of  the  calorimeter  and 
stirrer,  etc.,  t2  the  final  and  ti  the  initial  temperature  of  the 
calorimeter  and  water. 

Unless  special  precautions  are  taken  the  loss  of  heat  to  the  surroundings, 
which  is  largely  due  to  convection  (§323)  rather  than  radiation,  is 
relatively  great,  and  the  cooling  correction  is  a  very  important  one.  In 
general  it  can  be  reduced  by  protecting  from  air  currents,  polishing  the  er- 


230 


HEAT 


posed  surface  of  the  calorimeter,  surrounding  it  with  a  constant  tempera- 
ture enclosure,  and  arranging  matters  so  that  tl  and  tt  are  respectively 
slightly  below  and  above  the  temperature  of  the  enclosure.  In  some  cases 
it  is  impossible  or  inconvenient  to  use  water  as  the  calorimetric  substance, 
in  which  case  some  other  liquid  or  solid  of  known  specific  heat  may  be  used. 

288.  Application  of  Method  of  Mixtures.— The  method  of 
mixtures  may  be  used  for  different  purposes  according  to  the 
source  of  the  heat  H  which. is  to  be  measured.  One  important 
use  is  in  determining  the  specific  heat  of  substances.  For  this 
purpose  a  known  mass  M  of  the  substance  is  heated  to  a  tempera- 

TABLE  5 
SPECIFIC  HEATS 
(Calories  per  Degree  C.  per  Gram) 


Substance 

Specific  heats 

Temperature  C. 

Alcohol  (ethyl)  

0.548 

0° 

Aluminum  ,- 

0  219 

15  to  185 

Aluminum  ... 

0  0093 

—  240 

Brass  

0  090 

o 

Copper.  . 

0  0936 

20  to  100 

Copper  .  . 

0  00036 

—  250 

Diamond  

0  113 

11 

Diamond  

0.0003 

-220 

Glass  (flint)  

0  117 

10  to  50 

Gold  

0.0316 

0  to  100 

Granite  

0  19  to  0  20 

0  to  100 

Graphite  

0  160 

11 

Graphite  (acheson) 

0  0573 

-79  to  -190 

Ice  

0  502 

-21  to-1 

Iron  

0  119 

20  to  100 

Lead  

0  0305 

20  to  100 

Lead  

0  0143 

-250 

Mercury  

0  0333 

20 

Mercury  (solid) 

0  00329 

—  40  to  —75 

Nickel  

0  109 

18  to  100 

Platinum  

0.0323 

0  to  100 

Quartz  .  . 

0  174 

o 

Silver  

0  0559 

0  to  100 

Sodium  

0  2433 

—  83  to  —190 

Tin  

0  0552 

19  to  99 

Turpentine  

0  420 

18 

Sea  water  

0.980 

17 

Zinc  

0  0935 

0  to  100 

Zinc  

0.0017 

-240 

CALORIMETRY 


231 


ture  t  (above  or  below  ^)  and  added  to  the  calorimeter  and 
water.  The  temperature  of  the  calorimeter  and  of  the  mass  M 
will  then  equalize  and,  if  we  call  t2  the  final  temperature  of  the 
mixture,  the  heat  H  added  to  the  calorimeter  is  the  heat  lost  by 
the  mass  M  in  changing  from  the  temperature  t  to  t2,  which,  from 
the  definition  of  specific  heat,  is  equal  to  sM(t  —  t2)  if  s  is  the 


Water 


Fio.  190.  —  Continuous  flow  calorimeter  for  measuring  heat  of  combustion  of  gas. 

mean  specific  heat  of  the  substance  M  in  the  interval  t  to  Ja. 
then  have: 


We 


from  which  s  may  easily  be  computed. 

Sometimes  it  is  advisable  to  keep  the  hot  body  from  direct  con- 
tact with  the  water  by  putting  it  in  an  inner  vessel  having  thin 
walls  of  good  conducting  material. 

The  method  of  mixture  is  also  used  to  determine  heats  of  fusion 
(§305)  and  evaporation  (§312)  as  well  as  the  amount  of  heat 


232  HEAT 

developed  or  absorbed  in  various  chemical  reactions.  In  such 
cases  the  operation  consists  infusing,  or  condensing,  or  combining, 
as  the  case  may  be,  known  masses  of  material  inside  the  cal- 
orimeter (in  the  inner  vessel  above  referred  to),  and  special  forms 
of  calorimeters,  called  combustion  calorimeters,  bomb  calor- 
imeters, etc.,  have  been  developed  for  these  purposes. 

289.  Method   of  Continuous  Flow. — A   second   method   for   measuring 
quantities  of  heat  is  the  method  of  "  continuous  flow,"  illustrated  in  Fig.  190, 
in  which  a  steady  stream  of  the  calorimetric  substance  (usually  water  from  a 
reservoir)  at  a  constant  temperature  is  allowed  to  flow  past  the  point  at 
which  heat  is  being  set  free,  in  such  a  manner  that  all  of  the  heat  is  absorbed 
by  the  stream  of  water.    The  temperature  of  the  stream  of  water  is,  of 
course,  higher  after  the  heat  has  been  absorbed  than  before,  and  if  the  rate 
of  liberation  of  heat  is  constant  this  temperature  difference  will  be  con- 
stant  and  (neglecting  external  losses  as  before)  the  number  of  calories 
liberated  in  a  time  T  (since  it  does  nothing  but  heat  the  water)  will  be  equal  to 
the  number  of  grams  of  water  W  which  has  flowed  past  in  time  T,  multiplied 
by  the  number  of  degrees  rise  in  temperature  (<2  —  tj,  and  by  the  specific 
heat  of  water  s', 

or,    .  #  =  W(*a-*t) 

This  method  is  especially  useful  in  determining  the  heats  of  combustion  of 
gas  and  liquid  fuels  by  means  of  which  a  steady  rate  of  combustion  and 
hence  a  steady  liberation  of  heat  can  be  maintained.  This  method  can  also 
in  a  sense  be  reversed  by  generating  the  heat  mechanically  or  electrically, 
that  is,  by  converting  measured  amounts  of  mechanical  or  electrical  energy 
completely  into  heat,  which  is  absorbed  by  a  stream  of  fluid  whose  specific 
heat  is  to  be  determined.  The  above  equation  then  becomes: 

£F -!£«(*,— *0 

where  H  is  known  (from  mechanical  or  electrical  measurements)  in  energy 
units,  M  is  the  mass  of  fluid  flowing  past  in  time  T  and  s  is  its  specific  heat 
which  is  determined  by  this  equation  in  mechanical  units.  In  this  form  the 
method  has  been  used  by  Barnes  to  measure  the  specific  heat  of  water  and 
mercury,  and  it  is  capable  of  giving  very  accurate  results. 

290.  A  third  method  of  measuring  quantity  of  heat  is  the  "method  of 
latent  heats,"  in  one  form  of  which  the  heat  to  be  measured  is  used  to  melt 
a  measurable  amount  of  ice.     This  necessitates  a  knowledge  of  the  amount 
of  heat  required  to  melt  1  gram  of  ice  (heat  of  fusion  of  ice,  §305),  but 
has  the  advantage  that  the  calorimeter  remains  at  a  fixed  temperature,  0°  C. 
The  most  common  instrument  of  this  type  is  the  Bunsen  ice  calorimeter. 
A  second  form  is  the  Joly  steam  calorimeter,  in  which  the  order  of  tem- 
peratures is  reversed,  and  the  amount  of  heat  required  to  raise  M  grams  of 
a  substance  from  a  temperature  t  to  the  temperature  of  steam,  say  100°C., 
is  determined  from  the  weighed  amount  of  steam  which  is  condensed  to 


CALORIMETRY  233 

supply  this  heat.  A  knowledge  of  the  heat  liberated  in  condensing  1 
gram  of  steam  is,  of  course,  necessary.  This  is  a  very  convenient  and 
reliable  method. 

291.  The  Specific  Heat  of  Gases. — In  the  previous  discussion 
of  specific  heat  we  have  neglected  one  factor  which,  as  we  have 
seen  in  §279,  becomes  very  important  as  soon  as  we  consider 
gases,  namely,  the  expansion  which  usually  accompanies  rise  in 
temperature.  If  a  gas  is  confined  in  a  cylinder  with  a  movable 
piston,  as,  for  instance,  in  a  bulb  with  a  mercury  plug  in  an  attached 
capillary  tube,  Fig.  172,  and  is  heated,  it  will,  as  we  have  already 
noted,  expand  and  push  out  the  mercury  plug.  The  outside  of 
the  plug  is  acted  upon  by  the  pressure  of  the  air  which  opposes 
its  motion  outward  by  a  force  equal  to  the  product  of  the  pressure 
and  the  cross-section  of  the  tube.  Overcoming  this  force 
through  a  given  distance  means  doing  work,  called  the  external 
work  of  expansion,  and  this  work  has  evidently  been  done  by  the 
expanding  gas. 

Looking  at  the  matter  from  the  standpoint  of  the  kinetic  theory,  we  should 
say  that,  before  the  confined  gas  was  heated,  the  impact  of  gas  molecules  on 
the  inner  end  of  the  mercury  plug  (at  rest)  was  balanced  by  the  impact  of  air 
molecules  on  the  outer  end,  but  that  an  increase  in  temperature  of  the  gas 
meant  more  and  harder  impacts  on  the  inner  end,  thus  destroying  the 
equilibrium  and  causing  the  plug  to  move.  The  moving  plug  would,  on  the 
average,  hit  the  outside  molecules  harder  than  it  had  previously  done  when  at 
rest;  hence  it  would  increase  the  velocity  of  these  molecules  and  add  kinetic 
energy  to  them.  The  work  done  by  the  expansion  consists  in  a  transfer 
of  kinetic  energy  from  the  gas  molecules  inside  to  air  molecules  outside. 

Thus,  if  heat  energy  imparted  to  the  confined  gas  causes  ex- 
pansion some  energy  will  be,  by  this  expansion,  taken  out  of  the 
gas,  and  this  is  a  possible  disposition  of  part  of  the  added  energy 
quite  separate  from  those  considered  in  §262.  Hence  we  can  see 
that  to  raise  the  temperature  of  a  gas  with  the  volume  kept 
constant  must  take  an  amount  of  energy  different  from  that 
required  if  expansion  against  pressure  is  allowed,  not  only 
because  in  the  second  case  the  internal  potential  energy  may  be 
increased  but  because  external  work  is  done.  In  other  words, 
the  heat  added  to  a  gas  (or  any  body)  is  equal  to  the  increase  in 
internal  kinetic  and  potential  energy  plus  the  external  work  done. 

The  amount  both  of  the  internal  and  of  the  external  work  will 
evidently  depend  on  the  amount  of  expansion.  Since  the 


234 


HEAT 


increase  in  volume  of  gases  per  degree  rise  in  temperature  is  very 
much  greater  than  that  of  solids  or  liquids,  the  external  work  is 
greater.  If  the  volume  is  not  kept  constant  it  may  be  allowed 
to  vary  in  many  mays,  the  most  important  being  such  an  increase 
in  volume  that  the  pressure  remains  constant.  Hence  we  have 
the  specific  heat  of  a  gas  at  constant  volume,  sv,  and  the 
specific  heat  at  constant  pressure,  sp,  defined  as  the  heat  neces- 
sary to  raise  the  temperature  of  1  gram  of  the  gas  1°C.  under  the 
condition  of  constant  volume  or  constant  pressure  respectively. 
From  what  has  been  said  it  is  evident  that  sp  must  be  greater, 
in  general  considerably  greater,  than  sv. 

TABLE  6 
SPECIFIC  HEATS  OP  GASES  AND  VAPORS 


Substance 

Temperature 

Specific  heats 

Sp/SV 

Sp 

sv 

Alcohol  (ethyl) 

108-220 
0-100 
20-  90 
34-115 
15-100 
16-343 
27-118 
25-111 

.453 
.241 
.123 
.299 
.2025 
.113 
.144 
.428 

.400 

1.133 
1.402 
1.667 
1.397 
1.299 
1.336 
1.152 
1.024 
1.63 
1.408 
1.66 
1.41 
1.398 
1.33 

Air  

Argon  

.214 

Carbon  dioxide  

Chloroform  

.125 

Ethyl  ether  (C,H4)8O  

Helium  

Hydrogen  . 

12-198 
310 
0 
20-440 
100 

3.409 

Mercury  vapor  

Nitrogen 

.235 
.242 
.442 

Oxvcen  .  . 

Water  vapor 

The  measurement  of  s  has  been  most  accurately  made  by 
means  of  the  steam  calorimeter  (§290),  a  known  mass  of  the  gas 
being  enclosed  in  a  metallic  bulb,  and  the  weight  of  steam  con- 
densed in  raising  it  from  t°G.  to  100°G.  being  determined;  a 
correction  must  then  be  made  for  the  thermal  capacity  of  the 
bulb.  Sp  is  usually  measured  by  passing  a  stream  of  heated  gas 


CALORIMETRY 


235 


through  a  calorimeter.  According  to  Regnault  and  later  ob- 
servers sp  for  most  gases  varies  only  slightly  with  pressure,  while 
sp  for  air  is  almost  independent  of  the  temperature,  but  for  CO 
increases  very  markedly  with  temperature.  sv  for  air  and  C03 
increases  with  the  density  of  the  gas.  The  value  of  sv  has  not 
been  determined  directly  for  many  gases,  but  the  value  sp/sv  can 
be  readily  deduced  from  the  velocity  of  sound  in  the  gas  (  §  587). 

292.  The  Free  Expansion  of  a  Gas. — We  have  already  seen 
that,  if  there  are  forces  between  molecules  and  atoms,  when  a  gas 

expands  there  will  be  a  change  in  the 
potential  energy  of  its  molecules  (and 
perhaps  of  its  atoms),  since  the 
average  distance  between  molecules 
will  increase.  Work  done  against 
internal  forces  in  this  way  is  called 
the  internal  work  of  expansion  to 
distinguish  it  from  the  external  work 
done  against  the  pressure  confining 
the  gas. 

Gay-Lussac  and  later  Joule  at- 
tempted to  measure  the  internal 
work  by  the  method  of  free  expansion,  Fig.  191,  in  which  gas  was 
confined  at  some  considerable  pressure  in  the  vessel  A  and 
allowed  to  expand  quickly  through  a  cock  C  into  B  which  had 
been  highly  exhausted.  A,  B,  and  C  were  in  a  vessel  of  water 
whose  temperature  was  measured.  Since  expansion  occurred 
into  a  vacuum  it  was  "free"  (no  opposing  pressure)  and  hence, 
on  the  whole,  no  external  work  was  done;  but  if  there  were  any 
internal  work  done,  it  must  have  resulted  in  a  change  in  tempera- 
ture of  the  gas.  For  if  the  internal  potential  energy  increases, 
the  kinetic  energy  must  decrease  by  an  equal  amount,  that  is, 
the  temperature  of  the  gas  must  fall,  and  vice  versa.  Joule  did 
not  measure  the  temperature  of  the  gas  itself,  but  that  of  the 
water,  whose  large  heat  capacity  so  masked  the  effect  that  his 
results  merely  indicated  that  the  internal  work  of  expansion  is 
small. 

293.  Temperature  of  Gas  in  Motion. — If  a  gas  at  high  pressure 
and  ordinary  temperature  is  allowed  to  escape  into  the  atmos- 
phere through  a  fine  tube  (Fig.  192)  in  which  it  acquires  a  high 


Fio.  191. — Illustrating  Joule's  study 
of  the  "free  expansion"  of  gases. 


236 


HEAT 


velocity  of  flow,  very  marked  cooling  effects  will  be  observed 
where  the  velocity  of  flow  is  greatest,  though  the  total  energy  of 
the  moving  gas  is  practically  the  same  as  that  of  the  gas  at  rest. 
The  explanation  is  that  part  of  the  energy  of  the  random  undi- 
rected motion  of  the  particles  which  determines  the  temperature 
has  become  temporarily  energy  of  directed  motion  in  the  stream. 
But  if  the  gas  is  caught  in  a  large  receiver  and  allowed  to  come  to 
rest  its  temperature  will  be  found  to  be  slightly  higher  than 
before  expansion.  If  A,  B,  and  C  Fig.  191  are  placed  in  sepa- 
rate vessels  it  will  be  found  that  the  expansion  lowers  the  tempera- 
ture of  A,  and  raises  that  of  B  an  equal  a'mount.  The  reason 
for  this  is  that  the  gas  moving  out  of  A  corresponds,  with  respect 
to  the  gas  remaining  in  A}  to  the  piston  moving  away  from  the 
gas  in  Fig.  172,  hence  the  gas  remaining  does  work  on  the  gas 
which  is  going,  and  the  one  loses  and  the  other  gains  heat  of 
equal  amount. 


*•*- 


Fio.  192.  —  The  change  from  undirected  to  directed  molecular  motion  in  an  escaping  gas. 

294.  The  Difference  Between  the  Two  Specific  Heats.-—  From 
the  definition  of  sp  and  sv  in  §291  and  from  the  statements 
made  in  §292  we  see  that  if  we  denote  the  external  work  of 
expansion  by  We  and  the  internal  work  of  expansion  by  Wf 


The  expansion  is  that  necessary  to  maintain  P  constant  while 
t  rises  1°C.,  and  the  two  "works"  must  be  expressed  in  heat 
units.  For  gases  which  are  not  approaching  liquefaction,  that 
is,  for  example,  for  0,  H,  N,  and  air,  at  ordinary  temperatures, 
the  internal  work  of  expansion  is  so  small  that  it  may  usually  be 
neglected,  so  that  if  P  is  the  constant  pressure,  and  JV  the 


CALORIMETRY  237 

change  in  volume  per  degree  per  unit  mass,  and  /  the  mechanical 

PAV 
equivalent  of  heat,  then  the  external  work  =  —  — 

and  SP—  sv 

j 

Also,  PV  =  RT  approximately, 

and  P(V+JV)=R(T  +  1) 

Hence  PJ7  =  # 

R 

and  sp—  sv»»  -= 

j 

This  equation  was  used  by  Robert  Mayer  in  1842  to  make  the  first 
computation  for  J,  the  other  quantities  being  determined  by 
experiment. 

295.  The  Ratio  of  the  Two  Specific  Heats.  —  From  what  has  just  been'said, 
the  ratio  of  the  two  specific  heats  is  evidently: 

Wi    R 
SV+-T+J 

Sp  J       J 

Sy  Sy 

Also,  according  to  the  kinetic  theory,  sy=»  increase  in  molecular  kinetic 
energy  +  increase  in  atomic  energy,  per  degree;  of  which  the  first,  which 
we  shall  denote  by  Em,  represents  the  increase  in  temperature,  the  second 
the  increase  in  energy  inside  the  molecule  which  we  shall  represent  by  Ea. 
From  §227 


Hence 


so  that  Sy  = 


and 


3 

- 

Sp     2 


3 
2 


If  Ea  and  Wi  are  both  relatively  small,  as  we  should  expect  them  to  be 
simple  monatomic  gases  which  apppoximately  obey  Boyle's  law,  for  ex- 
ample, argon,  then 

Sp     5 

—  — -,  approximately. 

Sy     3 


238  HEAT 

On  the  other  hand  if  all  the  other  terms  are  negligible  compared  with  Ea< 
as  we  might  expect  for  gases  with  very  complicated  molecules,  for  example 
ether, 

8p 

—  =-1  approximately. 
Sy 

Both  results  are  in  agreement  with  the  values  given  in  §  291. 

296.  Expansion  Against  Pressure. — Let  a  gas  be  forced  through 
a  small  aperture  in  such  a  way  that  the  pressures  before  and  after 
passing  the  opening  are  maintained  constant.  A  possible  way 
of  doing  this  is  shown  in  Fig.  193,  in  which  the  pistons  both 
move  to  the  right  as  the  gas  passes  through,  and  the  external 
forces  upon  them  are  constant. 

Let  Pl   =  pressure  of  gas  before  expansion. 

Vl  =  specific  volume  before  expansion. 
P2  =  pressure  of  gas  after  expansion. 
V2  =  specific  volume  after  expansion. 


:\ 

'    ' 

Fj 

1 

p  ft/  \ 

F2 

'- 

'7V       ^/                                         1^- 

TO 

Fia.  193. — The    Joule-Kelvin   porous   plug   experiment.     Unbalanced   but   not   "free" 

expansion. 

Then  the  external  work  done  upon  the  gas  by  the  first  piston 
while  unit  mass  is  passing  is  PlVl  (§195),  the  external  work 
done  by  the  gas  after  expansion  upon  the  second  piston  is  P2Vz, 
and  (P1V1  —  P2V2)  'ls  ^ne  ne^  amount  of  external  work,  We,  done 
on  the  gas,  and  this  may  be  either  positive  or  negative.  The 
apparatus  is  supposed  to  be  so  made  that  no  heat  can  enter  or 
leave  the  gas  during  the  operation  and  the  temperature  of  the 
gas  is  observed  before  and  after  expansion. 

Let  W  i  again  represent  the  internal  work  done  against  the  forces 
between  molecules,  and  A£  the  observed  change  in  temperature 
of  the  gas.  Then  the  sum  of  the  external  and  internal  work  must 
be  equal  to  the  change  in  the  kinetic  energy  of  the  molecules  plus 
any  change  in  the  energy  inside  the  molecules,  and  this,  as  was 
just  shown  in  §295,  is  sy.  We  therefore  have  the  equation 


CALORIMETRY  239 

If  the  gas  strictly  obeyed  Boyle's  law  and  if  there  were  no  tem- 
perature change,  P^^  would  be  equal  to  P2V2,  that  is  We  would 
be  zero  and  hence,  by  the  above  equation,  the  internal  work 
would  be  zero.  As  a  matter  of  fact,  with  0,  N,  and  C02  a  cooling 
is  observed,  with  H  at  ordinary  temperatures  a  heating,  and  from 
these  observations,  combined  with  the  value  of  the  specific  heat 
sv  and  the  variation  of  P7,the  internal  work,  which  Joule  could 
not  detect,  can  be  computed.  The  results  indicate  in  all  cases 
molecular  forces  of  attraction.  To  avoid  false  cooling  effects  due 
to  mass  motion  of  the  gas  (§293) ,  it  is  customary,  following  the 
plan  of  Lord  Kelvin,  to  use  many  fine  openings — that  is  a  porous 
plug — hence  the  experiment  is  known  as  the  "porous  plug 
experiment." 

297.  Relations  between  Specific  Heats. — In  view  of  the  complicated  nature 
of  molecular  structure  as  already  outlined,  it  is  evident  that  no  simple 
relations  are  to  be  expected  between  the  specific  heats  of  different  substances 
or  of  the  same  substance  at  various  temperatures,  and  the  following  general 
statements  must  suffice. 

The  best  known  attempt  to  express  a  relation  between  the  specific  heats 
of  substances  is  the  so-called  "law"  of  Dulong  and  Petit  (1819),  which  states 
that  the  "product  of  the  specific  heat  by  the  atomic  weight  is  the  same  for  all 
solid  elements."  But  this  statement  and  its  extension  to  compound  sub- 
stances are  only  rough  approximations. 

In  general  the  specific  heat  of  substances  in  the  liquid  state  is  much  greater 
than  in  the  solid  (two  times  as  great  for  water,  ten  times  for  mercury), 
while  sp  for  the  gaseous  state  is  about  the  same  as  that  of  the  solid.  The 
change  of  specific  heat  from  solid  to  liquid  is  in  most  cases  smaller  with 
metals  than  with  non-metals. 

With  most  substances,  solid,  liquid  or  gaseous,  the  specific  heat  increases 
with  rise  of  temperature,  though  the  change  is  small  for  solids  with  the 
exception  of  carbon,  boron  and  silicon.  This  variation  of  the  specific  heat 
may,  according  to  §262,  be  due  to  several  causes,  such  as  an  increase  in 
the  relative  amount  of  kinetic  energy  inside  the  molecule  and  atom,  increase 
in  the  number  of  free  electrons,  or  an  increase  in  the  potential  energy  of 
molecular  groups  or  groups  of  atoms.  The  change  of  molecular  grouping  is 
probably  the  chief  cause  of  variation.  The  fact  of  variation  with  tempera- 
ture shows  at  once  that  Dulong  and  Petit's  "law"  cannot  be  a  general  one. 

Quite  recently  Nernst  has  extended  the  measurement  of  specific  heats 
down  to  23°  abs.  (  —  250°G.),  and  has  shown  that  the  specific  heat  of  all  the 
substances  examined  decreases  very  greatly  at  extreme  low  temperatures 
(Table  5).  He  has  also  developed  new  relations  between  the  specific 
heats  of  elements  and  compounds  which  promise  to  be  of  great  importance. 

298.  Heats  of  Combustion. — A  very  important  use  of  calorimeters  is  in 
measuring  heats  of  combustion  of  fuels,  that  is,  the  heat  liberated  by  the  burn- 


240 


HEAT 


ing  (in  air  or  oxygen)  of  1  gram  of  coal,  wood,  oil,  gas,  etc.  Such  fuels  are  the 
source  of  the  larger  part  of  the  available  energy  of  the  world,  and  a  knowl- 
edge of  the  energy  available  per  unit  mass  of  the  fuel  is,  of  course,  of  great 
importance  to  the  engineer.  The  method  of  mixtures  is  usually  used  for 
solid  fuels,  especially  with  one  form  of  apparatus  called  a  "  bomb  calorim- 
eter," in  which  a  weighed  amount  of  fuel  is  enclosed  with  compressed 
oxygen  in  a  steel  bomb  and  ignited  electrically,  the  bomb  being  in  the 
water  of  the  calorimeter.  For  liquid  fuels  and  gases  a  method  of  continu- 
ous flow  (§289)  is  also  very  much  used.  The  heat  of  combustion  is  usually 
expressed  in  calories  per  gram,  or  B.  T.  U.  per  pound  of  fuel. 

TABLE  7 
HEATS  OP  COMBUSTION.     (CALORIES  PER  GRAM.) 


Substance 

>. 

Substance. 

Alcohol  (ethyl) 

7183 

Gas  (coal  eras) 

5800-11000 

Alcohol  (methyl)  .  . 
Benzine 

5307 
9977 

Gas  (illuminating  gas).  .  . 
Gunpowder    

5200-5500 
730 

Carbon  (diamond) 

7860 

Hydrogen  

34100 

Carbon  (graphite).. 
Coal  (anthracite) 

7800 
7600-8400 

Petroleum  (Am.  crude).  . 
Wood  (beech)  

11100 
4168 

Coal  (bituminous) 

6100-7800 

Wood  (oak)  I 

3990 

Coal  (coke) 

7000 

Wood  (pine)  

4420 

CHANGE  OF  STATE 

299.  Change  of  State. — The  most  marked  changes  in  the 
physical  properties  of  bodies  occur  when  they  change  from  the 
solid  to  the  liquid  or  gaseous  state. 

The  change  from  the  solid  to  the  liquid  state  is  called  fusion 
or  melting,  the  reverse  change,  freezing. 

The  change  from  the  liquid  to  the  gaseous  state  is  called 
vaporization,  the  reverse  change  condensation. 

The  change  from  the  solid  directly  to  the  vapor  state  is  called 
sublimation,  the  reverse  change,  condensation. 

Each  of  these  changes  involves  a  rearrangement  of  the  mole- 
cules with  respect  to  each  other,  and  perhaps  a  rearrangement  of 
the  atoms  and  electrons  forming  the  molecules.  Vaporization 
and  sublimation  also  involve  a  very  great  increase  in  the  average 
distance  separating  molecules.  Rearranging  and  separating  the 
molecules  will  involve  an  increase  in  potential  energy  in  passing 


CHANGE  OF  STATE 


241 


from  the  solid  to  the  liquid  and  to  the  gaseous  state,  while  any 
change  in  volume  will  involve  doing  external  work  (§292) ; 
hence  energy  must  be  added  to  the  body  to  bring  about  the 
change.  Conversely,  when  a  vapor  condenses  or  a  liquid  solidi- 
fies, a  certain  amount  of  energy  must  be  taken  away  from  it. 
As  groups  of  liquid  molecules  "settle  down"  into  the  solid  ar- 
rangement, some  of  their  potential  energy  becomes  kinetic  and  is 
given  up  to  the  surface  on  which  freezing  occurs. 

300.  Fusion. — If    a   crystalline  solid  is  heated  while   acted 
upon  by  a  constant  pressure,  it  will  begin  to  melt  at  a  definite  tem- 
perature called  the  normal  fusing-,  or  melting-point,  and  the 
entire  mass  will  remain  at  this  temperature  until  it  is  all  melted. 
To  determine  this  temperature,  a  thermometer  bulb  of  some 
kind  protected  by  a  metal  or  porcelain  tube,  may  be  put  in  the 
mixture  of  solid  and  liquid,  as  in  Fig.  194.     Or,  a  thermometer 
may  be  placed  in  a  molten    substance  which    is    allowed    to 
slowly  lose  heat;  the  temperature  will  fall  until  solidification 
begins  after  whicli  it  will  remain  constant,  while  potential  energy 
(heat  of  fusion)  is  being  given  up,  until  solidification  is  complete. 
The  constant  temperature  is  the  freezing-point  of  the  substance 

The  freezing-point  of  water  is  found  by 
immersing  a  thermometer  in  a  mixture  of 
pure  ice  and  water,  carefully  protected 
from  gain  of  heat  from  the  outside.  As 
has  been  stated,  the  freezing-point  of  water 
under  one  atmosphere  pressure  is  one  of  the 
fixed  points  of  thermometry. 

301.  Effect  of  Pressure  on  Fusion. — The 
normal  melting-point  of  pure  substances 
depends  upon  the  pressure  under  which 
fusion  occurs — increase  of  pressure  raising 
the    melting-points    of    those    substances 
which  expand  on  melting  and  lowering  the 
melting-points  of  those  substances  which  contract  on  melting. 
This  relation  between  change  of  melting-point  with  pressure,  and 
change  of  volume  on  fusion,  was  deduced  first  from  theoretical 
considerations  by  James  Thomson.     The   curve   obtained   by 
plotting  melting-points  and  the  corresponding  pressures  on  the 
Pi  diagram  is  called  the  fusion-curve  (Fig.  200),  and  represents 

16 


Fia.  194.— Method  of  de- 
termining a  freezing-point 


242  HEAT 

the  conditions  under  which  the  solid  and  liquid  can  exist  in 
equilibrium  in  contact.  The  change  in  the  melting-point  with 
pressure  is  in  general  not  large,  being  for  water  0.0072°C.  decrease 
in  temperature  for  each  atmosphere  increase  in  pressure.  As  a 
result  of  the  work  of  Bridgman  and  Tammann  it  is  now  known 
that  water  can  exist  in  five  different  solid  forms,  which  can  be 
in  equilibrium  with  ordinary  ice  or  liquid  water  under  condi- 
tions roughly  given  by  the  dotted  curves  from  the  end  of  I, 
Fig.  200.  There  is  a  minimum  freezing  point  for  water  at 
—  22°C.  under  a  pressure  of  2500  atmospheres,  while  the  melt- 
ing-point of  one  form  of  ice  has  been  followed  up  to  80°C.  at  a 
pressure  of  20,000  atmospheres. 

TABLE  8 

TABLE  OF  MELTING-POINTS 
Subitance.  Melting-point.          Substance.  Melting-point. 

Aluminum 657°C.        Nickel 1452 

Copper 1083            Platinum....- 1753 

Gold 1063             Sliver 960 

Indium 2290            Tin.. 232 

Iron 1505            Tungsten 3270 

Lead 327            Zinc 418 

Mercury —38.8 

The  lowering  of  the  freezing-point  of  water  with  pressure  may 
be  strikingly  illustrated  and  has  important  consequences.  Two 
blocks  of  ice  at  about  0°C.  will  freeze  together  if  two  faces  are 
pressed  together;  snow  compressed  in  a  cylinder  becomes  a  clear 
transparent  mass  of  ice,  and  snow  at  about  0°C.  can  be  pressed 
by  the  hands  into  a  hard  snow  ball.  If  a  wire  supporting  a 
weight  is  looped  around  a  block  of  ice,  it  will  slowly  melt  its  way 
through  the  ice,  which  freezes  again  above  it.  A  further  illustra- 
tion is  the  fact,  well  known  to  skaters,  that  ice  is  more  slippery 
when  near  0°  than  when  many  degrees  below  zero.  In  all  these 
cases  the  pressure  applied,  which  may  be  very  considerable  at 
certain  points,  lowers  the  melting-point,  and,  if  the  initial  tem- 
perature of  the  snow  or  ice  is  not  too  low,  some  of  it  will  melt,  only 
to  freeze  again  when  the  pressure  is  relieved.  Thus  there  would 
be  a  film  of  water  between  the  skate  runner  and  the  ice.  If  the 
ice  or  snow  is  too  cold,  the  pressure  will  not  lower  the  melting- 
point  below  this  initial  temperature  and  no  melting  will  occur. 


CHANGE  OF  STATE 


243 


The  same  ideas  apply  to  the  "packing"  of  snow  on  roads,  and  on 
a  larger  scale  to  the  formation  of  glaciers  by  the  compression, 
melting,  and  regelation  of  snow  in  mountain  valleys.  The  sub- 
sequent flow  of  glaciers  down  the  valleys  is  due  in  part  to  the 
effect  here  discussed,  the  ice  melting  at  the  points  of  greatest 
pressure,  the  water  immediately  flowing  down  hill  a  little,  thus 
relieving  the  pressure,  and  then  freezing  again. 

302.  Crystalline  and  Amorphous  Solids. — The  sharp  change 
from  solid  to  liquid  at  a  definite  temperature,  which  we  have  been 
discussing,  is  characteristic  of  solids  which  have  a  definite 
crystalline  structure.  Solids  which  have  not  such  a  structure, 


ifoU 
326 

<t 

Fret 

zing  Point 

Guru 

^ 

250 

201) 
1H> 

^ 

X< 

L 

md-1 

in  All 

zys 

>s 

^k 

^ 

to^. 

-•^ 

^-^*" 

Eutectic 

0        10 


20 


30        40        50       60        70 

PERCENTAGES  OF  TIN  BY  WEIGHT 


90      100 


FIG.  195. — Freezing-points  of  alloys.     Upper  line,  beginning  of  freezing;  lower  line  end  of 
freezing;  eutectic  has  a  sharp  freezing-point. 

called  amorphous  solids,  of  which  fats,  waxes,  glass  and  most 
alloys  are  examples,  change  gradually  from  one  state  to  another, 
that  is,  gradually  soften  throughout  the  entire  mass,  while  the 
temperature  rises  slightly,  there  being  no  definite  "melting- 
point."  Amorphous  solids  are  in  general  mixtures.  Some 
alloys,  however,  are  definite  compounds,  having  marked  crystal- 
line structure  and  very  definite  melting-points.  The  freezing- 
point  curve  for  a  simple  group  of  alloys  is  shown  in  Fig.  195. 

303.  Change  of  Volume  on  Freezing. — Most  substances  con- 
tract on  freezing,  the  solid  sinking  in  the  liquid.  The  fact  that 
iron,  bismuth  and  antimony  and  some  alloys  such  as  type-metal 
(lead,  antimony,  and  tin)  expand  on  solidifying  is  valuable 
industrially,  since,  when  cast,  they  take  a  particularly  sharp 
impression  of  the  mold.  The  expansion  of  water  upon  freezing 
is  responsible  for  the  bursting  of  water  pipes,  the  bursting  (and 
hence  death)  of  plant  cells,  and  the  splitting  of  trees  and  rocks. 
Very  carefully  dried  seeds  may  be  put  in  liquid  air  without  injury, 


244  HEAT 

but  the  presence  of  the  slightest  trace  of  moisture  will  result  in 
killing  the  seeds. 

304.  Freezing  Point  of  Solutions. — The  fact  that  a  dilute  solu- 
tion, such  as  sea  water,  has  a  lower  freezing-point  than  the  pure 
solvent,  and  that  the  lowering  of  the  freezing-point  of  dilute 
solutions  is  approximately  proportional  to  the  amount  of  sub- 
stance dissolved  has  been  known  for  a  long  time.     The  depression 
of  the  freezing-point  per  gram-molecule  of  salt  dissolved  in  100 
grams  of  solvent,  calculated  from  observations  on  dilute  solutions, 
is  called  the  molecular  lowering  of  the  freezing-point.     Later  work 
shows  that  the  freezing-point  of  a  given  solvent  is  lowered  the 
same  amount  by  many  different  salts  when  dissolved  in  propor- 
tion to  their  molecular  weights,  while  other  salts  will  produce  a 
depression  two  or  three  times  as  great. 

According  to  the  dissociation  hypothesis,  abnormally  large  depressions 
are  due  to  the  breaking  up,  or  dissociation,  of  molecules  into  parts,  while 
abnormally  small  depressions  are  due  to  the  grouping  together  of  molecules. 
Thus  common  salt  in  water  apparently  dissociates  into  Na  and  Cl  giving  a 
solution  which  conducts  electricity  readily,  and  producing  a  molecular 
lowering  of  the  freezing  point  of  about  3.6°.  If  the  temperature  of  a  given 
dilute  solution  is  lowered  beyond  the  freezing  point  corresponding  to  its 
saturation,  the  pure  -solvent  only  will  begin  to  freeze  out  of  the  solution, 
which  becomes,  therefore,  more  concentrated,  until,  on  continued  cooling,  a 
certain  definite  concentration  is  reached  (depending  upon  the  pressure) 
when  the  entire  mass  freezes  as  a  mixture  of  the  two  solids.  This  mixture 
is  called  a  cryohydrate^  The  corresponding  mixture  in  case  of  alloys,  having 
a  minimum  melting-point  as  compared  with  other  percentage  compositions, 
is  called  an  eutectic.  (Fig.  195.) 

305.  Heat  of  Fusion. — The  heat  of  fusion  of  any  substance  is 
defined  as  the  number  of  calories  required  to  convert  one  gram  of  the 
solid  at  the  melting-point  into    liquid  at  the  same  temperature. 
Heats  of  fusion  are  usually  measured  by  some  modification  of 
the  method  of  mixtures. 

Thus  if        M  =  no.  of  grams  of  melted  substance  used, 

£s  =  temperature  of  substance  when  added  to  calo- 
rimeter, 

tm  =  melting  point  of  substance, 
£2  =  final  temperature  of  calorimeter, 
ti  «=  initial  temperature  of  calorimeter, 
w  =  mass  of  water  used, 
Si=heat  capacity  of  calorimeter, 


CHANGE  OF  STATE 


245 


Sj  =  specific  heat  of  substance  when  melted, 
s,  =  specific  heat  of  substance  when  solid, 
L  =  heat  of  fusion, 


then 


from  which  L  may  be  computed. 

TABLE  9 
HEATS  OP  FUSION 
(Calories  per  gram) 

Aluminum  ...............................  77 

Copper  ..................................  43 

Ice  ......................................  79.8 

Lead  ....................................  5 

Mercury  .................................  3 

Platinum  ................................  27 

Sulphur  .................................  9 

Zinc  ....................................  28 

306.  Vaporization.  —  From  the  molecular  standpoint,  vaporiza- 
tion means  the  flying  off  of  molecules  against  the  forees  of  molecu- 
lar attraction,  these  molecules  losing  kinetic  energy  and  gaining 
potential  energy  as  they  leave  the  liquid.  The  more  rapidly 
moving  molecules  will  be  the  first  to  fly  off,  hence  the  average 
kinetic  energy  of  the  molecules  remaining  behind  will  be  less  than 
the  initial  average  for  the  liquid,  and  the  liquid  will  be  cooled  by 
evaporation.  If  the  vapor  is  confined  over  the  liquid,  some  vapor 
molecules  will  strike  the 
surface  and  become  liquid 
again,  and  as  the  number 
of  vapor  molecules  per 
unit  volume  (i.e.,  the 
density  of  the  vapor)  in- 
creases, the  number  of 
molecules  returning  to 
the  liquid  per  second  will 
likewise  increase,  until 
finally  the  average  num- 

ber  returning   will  equal          Fia-  196.—  Pressure  (tension)  of  water  vapor  at 
.  various  temperatures. 

the      average      number 

leaving.     Under  these  conditions  the  vapor  is  in  equilibrium  with 

the  liquid.   The  density,  and  hence  the  pressure,  of  the  vapor  which 


1 

N  CENTIMETERS  OF  HO. 

k  »—  *  tO  CO 

1  8  1  § 

1 

I 

( 

•n 

-NJ 

10 

^  0 

F\ 

VA 

FEI 

*  V 

AP 

DR 

/ 

( 

/ 

/ 

TENSION 
08? 

/ 

f 

/ 

r 

LX 

i  —  r 

—  —  ' 

- 

—  * 

40        60 


80      100 

TEMPERATURES 


120      140    160 


246 


HEAT 


will  be  in  equilibrium  will  depend  on  the  temperature,  that  is,  on 
the  average  molecular  velocity.  A  vapor  in  equilibrium  with 
the  liquid  is  said  to  be  saturated,  and  the  equilibrium  pressure  is 
called  the  saturated  vapor  pressure  (or  vapor  tension),  which  for  a 
given  substance  depends  only  upon  the  temperature.  If  the 
vapor  is  not  allowed  to  accumulate  over  the  liquid  it  will  remain 
unsaturated,  equilibrium  will  not  be  reached,  and  the  liquid  will 
gradually  disappear  by  evaporation. 

TABLE  10 
VAPOB  TENSIONS  AND  VAPOR  DENSITIES  OP  WATER 


Temperature. 

Vapor  tensions 
(mm.  of  Hg.). 

Densities  of  saturated 
vapor  (grams  of  vapor  per 
cu.  m.  of  saturated  air). 

-20°  C. 

0.781 

0.892 

-10 

1.961 

2.154 

0 

4.579 

4.835 

10 

9.205 

9.330 

20 

17.51 

17.118 

30    • 

31.71 

30.039 

40. 

55.13 

50.625 

50 

92  30 

60 

149  2 

70 

233.5 

80 

355  1 

90 

525.8 

100 

760 

140 

2709 

180 

7514 

260 

35760 

360 

141870 

No  general  relation  is  known  connecting  the  saturated  vapor 
pressure  and  temperature,  though  many  empirical  relations  have 
been  found  which  are  satisfactory  in  certain  cases.  The  corre- 
sponding values  of  temperature  and  saturated  vapor  pressure 
for  water  are  shown  in  Table  10  and  Fig.  196.  Points  in  this 
diagram  indicate  the  physical  condition  of  water  substance; 
points  on  the  curve  show  the  conditions  under  which  water  may 
exist  either  as  a  vapor  or  liquid  or  both  in  equilibrium,  as,  for 
example,  at  a  temperature  of  140°C.  and  under  a  pressure  of  270 
cm.  of  mercury.  If  the  pressure  is  increased,  without  suitably 
raising  the  temperature  so  as  to  reach  another  point  on  the  curve, 


CHANGE  OF  STATE 


247 


all  the  vapor  will  be  condensed,  while  if  the  temperature  is  in- 
creased without  properly  increasing  the  pressure,  all  the  water 
will  vaporize.  Hence  this  curve,  which  represents  equilibrium 
conditions,  divides  other  conditions  into  two  groups,  an  all- 
vapor  group  represented  by  points  to  the  right  of  and  below  the 
curve,  and  an  all-liquid  group  represented  by  points  to  the  left 
of  and  above  the  curve. 

307.  Humidity. — The  saturated  vapor  pressure  for  a  given 
temperature  is  not  measurably  affected  by  the  presence  of  gases 
which  do  not  chemically  combine  with  the  vapor.  When  we 
speak  of  air  being  saturated  with  water  vapor,  what  we  really 
mean  is  that  the  vapor  is  saturated.  The  presence  of  air  above 
a  water  surface  will  not  influence  the  vapor  pressure  necessary 
for  equilibrium,  but  will  slightly  influence  the  rate  of  evaporation 
if  the  equilibrium  condition  is  not  reached. 

The  degree  of  saturation  of  air  with  water  vapor  is  of  great 
importance  in  its  influence  upon  climate,  for  it  determines  the 
rate  at  which  evaporation  will  go  on 
from  exposed  surfaces  of  water  or  from 
moist  surfaces,  such  as  that  of  the 
human  body.  Evaporation,  as  we  have 
seen,  causes  cooling;  hence  the  less 
saturated  the  air  the  greater  the  cool- 
ing, since  evaporation  will  be  more 
rapid.  Thus  a  given  summer  tempera- 
ture with  the  air  dry  is  less  oppressive 
than  with  the  air  nearly  saturated . 
The  effective  dryness  of  air  depends  on 
its  degree  of  saturation,  and  this  is 
called  the  humidity,  absolute  humidity 
being  denned  as  the  mass  of  water 
vapor  contained  in  a  cubic  centimeter 
of  air  at  a  given  temperature,  and  rela- 
tive humidity  as  the  ratio  of  the  mass 
of  moisture  actually  present  to  the 
amount  needed  for  saturation. 

If  water  vapor  (or  air  and  water  vapor)  is  heated  at  approxi- 
mately constant  pressure,  without  the  addition  of  vapor,  as  in 
a  hot  air  furnace,  it  expands,  and  therefore  the  mass  of  vapor 


9 

a 

» 

-120 

1 

-120 

-110 

110 

-100 

-TOO 

-90 

-90 

-80 

-80 

-70 

-70 

-60 

'-60 

'-50 

-SO 

-40 

-40 

-SO 

-80 

-20 

-20 

-10 

-10 

-0 

-  0 

-10 

-10 

-20 

-20 
-30 

FIG.  197.— Wet  and  dry  bulb 
.          hygrometer. 


248 


HEAT 


per  unit  volume  decreases.  At  the  higher  temperature  the 
density  necessary  for  saturation  is  greater,  however,  so  that  for 
two  reasons  the  effective  dryness  is  increased.  As  might  be  ex- 
pected, the  air  in  houses  in  winter  is  usually  far  too  dry  for  either 
comfort  or  good  health. 

The  measurement  of  humidity  is  called  hygrometry.  The  wet 
and  dry  bulb  hygrometer  (Fig.  197)  consists  of  two  exactly  similar 
thermometers  similarly  exposed,  except  that  the  bulb  of  one  is 
covered  with  a  light  wick  kept  moist  by  dipping  in  a  vessel  of 
water.  Evaporation  from  the  wet  bulb  will  cool  it  as  compared 
with  the  other,  and  the  dryer  the  air  the  greater  will  be  the 
difference  in  temperature  between  the  two.  By  noting  this 
.difference  and  the  temperature  of  the  dry  bulb,  either  the  relative 
or  the  absolute  humidity  may  be  obtained  from  tables.  Air 
must  circulate  freely  around  these  thermometers  in  order  that 
they  should  give  accurate  results. 

Much  more  reliable  results  are  obtained  if  the  wet  and  dry  bulb  thermom- 
eters are  swung  rapidly  through  the  air,  this  form  being  called  the  "sling 

psychrometer."  Other  methods  depend  upon 
the  determination  of  the  dew-point,  that  is, 
tne  temperature  at  which  air  would  be  satur- 
ated by  the  moisture  actually  in  it.  One 
form  of  dew-point  hygrometer,  Regnault's,  is 
shown  in  Fig.  198.  The  central  chamber  into 
which  the  thermometer  dips  is  cooled  by  the 
evaporation  of  the  ether  contained  in  it,  and  the 
temperature  at  which  condensation  first  occurs 
on  the  front  polished  metal  face  is  the  dew- 
point.  At  ordinary  room  temperatures],  the 
proper  relative  humidity  is  from  60  to  70  per 
cent.  To  reach  this  humidity,  it  is  necessary 
to  evaporate  water  in  the  furnace  or  in  the 
rooms  themselves.  If  evaporation  is  desired 
'in  the  furnace,  it  is  evident,  from  what  has 
been  said  above,  that  the  hot  air  (not  the  cold 
air  as  is  usually  the  case)  should  pass  over  the 

water  surface  in  order  to  take  up  as  much  water  vapor  as  possible  and  not 
be  "  dried  "  by  subsequent  heating.  To  give  70  per  cent,  relative  humidity  at 
68°F.  in  a  house  40  ft.  bjf  30  ft.  and  25  ft.  high,  containing  about  30,000  cu. 
ft.K  requires  about  22|  Ibs.  or  2.8  gallons  of  water,  and  usually  several  times 
this  amount  must  be  supplied  per  day  to  maintain  proper  conditions. 

308.  Boiling-point. — The  boiling-point  of  a  liquid  is  the  temper- 
ature at  which  its  saturated  vapor  pressure  is  equal  to  the  atmos- 


nHHh^0  Aspirator 


Polished 
Gold  Surfaces 


Fio.  198.— Regnault's  dew- 
point  hygrometer. 


CHANGE  OF  STATE  249 

pheric  pressure  on  the  liquid  surface.  At  this  temperature, 
bubbles  of  vapor  will  form  in  the  liquid  and  escape  through  the 
surface,  and  this  formation  and  escape  of  vapor  bubbles  is  called 
boiling.  Evidently  the  boiling-point  will  be  higher  the  greater 
the  pressure  on  the  surface  of  the  liquid,  and  the  variation  of  the 
boiling-point  with  pressure  is  the  same  thing  as  the  change  of 
saturated  vapor  pressure  with  temperature.  In  the  determina- 
tion of  boiling-points  the  thermometer  is  put  in  the  vapor  rather 
than  in  the  liquid,  and  it  must  be  protected  from  gain  or  loss  of 
heat  by  radiation  and  from  the  condensation  of  liquid  upon  it. 

TABLE  11 

BOILING-POINTS  UNDER  ONE  ATMOSPHERE  PRESSURE 
Substance.  Boiling-points. 

Alcohol  (ethyl) 78.3 

Benzol ^ 80.2 

Carbon  dioxide —  78 . 2 

Chloroform 61 . 2 

Ether  (ethyl) 34.6 

Iron 2450 

Mercury 357 

Oxygen  (liquid) — 182.9 

Pentane 36.2 

309.  Effect  of  Pressure  on  Boiling-point. — The  variation  of 
the  boiling-point  with  pressure  may  be  determined  by  enclosing 
the  liquid  to  be  boiled,  reducing  the  air  pressure  on  the  surface 
and  noting  the  temperature  at  which  steady  boiling  takes  place. 
This  may  be  done  by  placing  the  flask  under  the  receiver  of  an 
air  pump.  If  the  space  over  the  liquid  contains  only  the  satu- 
rated vapor,  and  the  pressure  of  this  is  suddenly  reduced,  as  for 
example,  by  pouring  cold  water  over  a  sealed  flask  containing 

TABLE  12 

CHANGE  OP  BOILING-POINT  OP  WATER  WITH  PRESSURE 
Pressure  Pressure 

(in  mm.  Hg.).        Boiling-point.  (in  mm.  Hg.).        Boiling-point. 

680  96.91°C.  740  99.25 

690  97.32  750  99.63 

700  97.71  760  100 

710  98.11  770  100.37 

720  98.49  780  100.73 

730  98.88 


250  HEAT 

water  and  its  vapor  slightly  below  the  boiling  point,  boiling  will 
at  once  begin.  The  pressure  variation  of  the  boiling-point  of 
water  is  frequently  used  to  determine  the  air  pressure  on  moun- 
tain tops,  and  hence  roughly  their  height.  The  variation  for 
water  is  about  0.37°C.  at  100°  for  a  change  in  pressure  of  1  cm. 
of  mercury. 

310.  Other   Conditions  Affecting  Boiling. — The  ease  with  which  vapor 
bubbles  are  formed  in  a  liquid  depends  upon  various  conditions,  such  as  the 
presence  of  dissolved  gases,  or  of  points  or  small  solid  particles  in  the  liquid. 
Hence  the  prevention  of  "  bumping, "  or  the  violent  formation  of  vapor 
bubbles,  is  brought  about  by  putting  broken  glass  or  other  rough  solids  into 
the  boiling  liquid.     The  boiling-point  of  a  given  pure  liquid  is  always  raised 
by  dissolving  any  relatively  non-volatile  substance  in  it,  as,  for  example, 
sugar  in  water,  but  may  be  either  raised  or  lowered  by  dissolving  a  volatile 
substance  in  it,  as,  for  example,  alcohol  in  water,  which  gives  solutions 
boiling   below  the  boiling-point   of    water,    but    above   that  of  alcohol. 
With  volatile  combinations   the   boiling  point    may  be  either  above  or 
below  the   boiling  point    of   both   constituents,  while  with  a  non-volatile 
solute  the  elevation  of  the  boiling  point  is,  for  dilute  solutions,  propor- 
tional to  the  mass  of    solute  added  and  approximately  the  same  for 
equal  gram-molecular  weights  of  all  solutes  in  the  same  amount  of  &  given 
solvent.    For  water  the  elevation  is  at  the  rate  of  5°  for  each  gram-molecular 
weight  dissolved  in  100  grams  of  water,  and  this  is  called  the  molecular  rise 
of  the  boiling-point. 

311.  Specific  Volumes  and  Densities  of  Vapors. — In  general 
there  is  a  very  large  increase  in  volume  on  vaporization.     Thus 
1  gram  of  saturated  steam  at  100°C.  occupies  1721  c.c.  while  1 
gram  of  water  at  100°G.  occupies  only  1.043  c.c.     The  volume 
of  1  gram  of  a  substance  is  termed  its  specific  volume,  and  is 

evidently  equal  to  -,  where  p  is  the  density.     As  the  temperature 

is  raised  the  specific  volume  of  any  saturated  vapor  decreases, 
whereas  that  of  the  liquid  increases.  Fig.  199  shows  this  be- 
havior for  water  and  steam.  The  two  specific  volumes  of  liquid 
and  vapor  become  equal  at  a  definite  temperature,  called  the 
critical  temperature,  which  will  be  further  discussed  in  §316. 

312.  Heat  of  Vaporization. — The  heat  of  vaporization  is  the 
number  of  calories  required  to  change  1  gram  of  a  substance  from 
a  liquid  at  a  certain  temperature  to  a  vapor  at  the  same  temperature 
under  a  specified  pressure,  the  symbol  being  Lt.     The  method  of 
mixtures  is  usually  used  to  determine  the  heat  of  vaporization, 
vapor  from  a  boiling  liquid  being  passed  into  a  vessel  immersed 


CHANGE  OF  STATE 


251 


in  the  water  of  a  calorimeter,  where  the  vapor  is  condensed  and 
the  heat  given  off.  If  M  grams  of  vapor  at  a  temperature  ta 
in  condensing  raise  m  grams  of  water  and  the  calorimeter  (heat 
capacity  S^HS^  from  tt  to  the  final  temperature  tz  of  the  mixture, 


from  which  Lt  can  be  computed,  if  s  the  specific  heat  of  the 
condensed  vapor,  is  known. 

Since  there  is  in  general  a  very  considerable  increase  in  volume 
on  vaporization,  which  occurs  against  a  definite  pressure,  external 
work  must  be  done  by  the  vapor  as  it  is  formed.  The  heat  of 


40 


£30 

£ 

I 

§20 


10 


Specific  Volume 
Water  Substarice 


Liquid 


0         100         200         300      374  400 

TEMPERATURES 

FIG.  199. — Specific  volumes  of  liquid  and  saturated  vapor  at  various  temperatures. 


vaporization  may,  therefore,  be  considered  as  made  up  of  two 
parts,  the  internal,  which  is  the  increase  in  potential  energy  of 
molecules  and  atoms,  and  the  external,  which  is  the  work  done  by 
expansion. 

The  heat  of  vaporization  diminishes  with  increasing  tempera- 
ture and  becomes  zero  at  the  critical  temperature,  where,  as  we 
shall  see  (§318),  the  distinction  between  liquid  and  vapor  van- 
ishes. The  heat  required  to  change  1  gram  of  a  substance  from 


252  HEAT 

a  liquid  at  0°C.  to  a  vapor  at  a  temperature  t  is  called  the  total 
heat  of  the  vapor  at  temperature  t.  If  Ht  is  the  total  heat,  and 
we  assume  the  specific  heat  is  independent  of  temperature, 

Ht=st+Lt 

To  supply  this  heat  either  the  remaining  liquid  will  be  cooled,  or  heat 
will  be  drawn  from  the  surroundings — hence  the  cooling  effect  of  evapora- 
tion. The  rate  of  evaporation  will  depend  upon  the  rate  of  supply  of  heat; 
hence  boiling  over  a  fire  will  be  more  violent  in  a  metal  than  in  a  non-con- 
ducting vessel.  It  is  the  heat  of  vaporization  of  steam  which  is  chiefly 
effective  in  steam  heating  systems,  this  heat  being  supplied  in  the  boiler  and 
given  up  (potential  converted  to  kinetic  energy  again)  in  the  radiators  where 
the  steam  is  condensed.  The  use  of  a  hand  or  electric  fan  in  warm  weather 
is  the  most  common  example  of  the  reverse  process,  that  is,  cooling  resulting 
from  increased  evaporation  due  to  the  circulation  of  the  air. 

TABLE  13 

HEAT  OF  VAPORIZATION 
(Calories  per  gram  at  normal  boiling  points.) 

Alcohol  (ethyl) 205      Liquid  air 50 

Liquid  H, 123      Liquid  COa 96 

Liquid  Oa 58      Mercury 68 

Liquid  N8 50      Water 542 

313.  Sublimation. — The  direct  change  from  the  solid  to  the 
vapor    state    is    termed    sublimation.     This*    or    the    reverse, 
direct  condensation,  is  commonly  observed  in  the  evaporation  of 
snow  in  cold  dry  weather,  in  the  production  of  hoar  frost,  and  in 
the  evaporation  and  recondensation  of  camphor  confined  in  a 
bottle.     As  in  the  other  two  transformations,  there  is  a  definite 
vapor  pressure,  for  every  temperature,  at  which  the  solid  and 
vapor  can  exist  in  equilibrium  together,  without  one  state  con- 
tinuously changing  into  the  other.     When  plotted  on  the  Pt 
plane  these  points  give  the  "hoar  frost"  or  sublimation  line,  the 
equilibrium  pressure  falling  with  the  temperature.     Sublimation 
also  involves  an  increase  in  potential  energy  and  external  work, 
and  hence  a  heat  of  sublimation. 

314.  Unstable  Conditions. — In  stating  that  substances  solidify, 
vaporize,  and  condense  at  definite  temperatures  under  a  given 
pressure,  we  have  disregarded  certain  cases  in  which  it  is  possible 
to  cool  a  liquid  very  much  below  its  freezing  point  without  solidi- 
fication and  to  heat  a  liquid  very  much  above  its  boiling-point 
without    vaporization.     These,    however,    are    abnormal    and 


CHANGE  OF  STATE 


253 


unstable  conditions,  since,  if  freezing  (or  boiling)  once  starts,  it 
goes  on  with  great  violence  till  the  normal  temperature  has  been 
restored.  For  example,  small  drops  of  water  in  an  oil  of  equal 
density  have  been  cooled  at  atmospheric  pressure  to  —  20°C 
without  freezing  and  heated  to  178°C.  without  boiling,  and 
minute  spheres  of  platinum  and  other  metals  have  been  cooled 
several  hundred  degrees  below  their  normal  melting  point  before 
solidification  occurred.  Agitation,  the  presence  of  points  or 
particles,  or  the  least  trace  of  the  solid  serve  to  start  solidification, 
while  points  and  pieces  of  porous  solids  start  boiling.  A  liquid 
above  its  boiling-point  will  begin  to  boil  violently  if  touched  by  a 
file  or  paper — though  these  materials  become  ineffectual  when 
they  become  clean. 

Condensation  of  a  vapor  is  difficult  to  start  without  the  presence  of  nuclei, 
that  is,  dust  particles,  liquid  droplets  or  electrified  molecules  or  atoms,  called 
tons,  which  "very  greatly  assist  the  formation  of  large  drops.  The  efficacy 
of  some  of  these  is  due  to  their  providing  surfaces  of  relatively  larger  radius 
on  which  condensation  can  take  place,  since  the  vapor  pressure  necessary 
or  the  equilibrium  of  a  vapor  with  a  liquid  drop  is  less  for  a  drop  of  large 
radius,  and  also  less  for  electrified  drops. 

315.  The  Triple  Point  Diagram. — Having  discussed  the  three 
equilibrium  curves,  solid-liquid,  liquid-vapor,  and  solid-vapor, 
let  us  consider  them  combined 
as  in  Fig.  200  in  the  Pt  plane. 
Since  the  areas  on  either  side  of 
each  cuver  represent  conditions 
such  that  only  one  state  can 
exist,  for  example  solid  to  the 
left  and  liquid  to  the  right  of 
the  freezing-point  curve,  a  little 
consideration  will  show  that,  in 
order  to  be  consistent,  the  three 
curves  must  intersect  in  a  point. 
If  this  were  not  so  the  area 
included  between  the  curves 
would  represent  quite  contra- 
dictory conditions  as  deduced 
from  the  several  curves.  Since  each  curve  represents  conditions 
of  possible  coexistence  of  two  states,  in  the  condition  repre- 
sented by  the  point  of  intersection  all  three,  solid,  liquid,  and 


Gas 


Schematic 

Triple  Point  Diagram 
For  Water 


TEMPERATURES 


20     0     20    40    60     80 
Fio.  200.— Triple  point  diagram. 


254  HEAT 

vapor  can  exist  simultaneously  in  equilibrium  and  this  is  the 
only  condition  in  which  this  is  possible.  It  is  called  the  triple 
point  and  for  water  corresponds  to  -f-0.0072°C.,  and  a  pressure 
of  about  4.6  mm.  of  mercury. 

It  may  be  thought  that  common  experience  contradicts  this  conclusion 
as  to  the  unique  properties  of  the  triple  point — ice,  water,  and  water  vapor 
being  often  found  coexistent  at  various  temperatures.  Under  these 
conditions  it  will  be  found,  however,  that  the  ice  is  melting  and  the  vapor 
condensing,  or  water  evaporating — at  least  equilibrium  does  not  exist.  The 
characteristics  represented  by  the  triple-point  diagram  are  true  for  all 

substances. 

• 

316.  The  Critical  Temperature. — In  1822  Cagniard  de  la  Tour 
made  important  discoveries  regarding  the  relation  between  the 
liquid  and  the  vapor  state  by  filling  a  glass  tube  with  alcohol  and 
its  vapor,  sealing  off  the  tube  and  heating  it.     If  about  two-thirds 
of  the  total  volume  were  liquid  at  ordinary  temperatures  he 
found  that,  as  the  temperature  increased,  the  meniscus,  or  curved 
surface  separating    liquid  from  vapor  and  due  to  molecular 
attractions  (§208),  became  flatter  and  less  distinct  and  finally 
disappeared.     Thus    above    a    temperature    of    about   243°C. 
the  liquid  and  vapor  have  the  same  molecular  attractions  (no 
meniscus)  and  are  visually  identical. 

The  limiting  temperature  at  which  a  separation  can  be  observed 
between  the  liquid  and  the  vapor  state  is  called  the  critical  tempera- 
ture. We  have  already  seen  that  the  specific  volumes  of  a 
liquid  and  its  vapor  become  more  nearly  equal  as  the  temperature 
is  raised,  and  that  at  the  same  time  the  latent  heat  of  vapori- 
zation becomes  less,  and  the  relation  of  these  facts  will  now 
be  clear. 

317.  Isothermal  Curves  for  C02. — Andrews,  in  1863,  inclosed 
G02  in  a  glass  tube,  kept  this  at  a  constant  temperature  and  com- 
pressed the  gas  by  forcing  mercury  into  the  tube.     He  measured 
the  pressure  required  to  compress  the  gas  to  various  measured 
volumes  while  tke  temperature  was  kept  constant,  and  did  this 
at  a  series  of  temperatures  from  13°  to  48°G.  Corresponding 
pressures  and  volumes  plotted  on  the  PV  plane  gave  what  we  have 
called  isothermal  curves,  shown  in  Fig.  201. 

The  effect  of  compression  at  21.5°  from  an  initial  volume  of  12 
c.c.  per  gram  is,  as  shown  by  the  curve,  first  a  gradual  increase  in 
pressure  until  a  pressure  of  59  atmospheres  is  reached  (at  A) 


CHANGE  OF  STATE 


255 


when  liquid  C02  will  suddenly  appear  in  the  tube,  after  which 
no  increase  in  pressure  will  occur  in  spite  of  diminution  in  volume 
until  B  is  reached.  During  this  time  condensation  has  continued, 
until  (at  B)  the  vapor  has  been  entirely  changed  to  liquid,  after 


?  4  5  6  7  8  9 

Cubic  Centimeters  per  Gram 

Fia.  201. — Isothermals  of  carbon  dioxide. 


11         12 


which  any  decrease  in  volume  necessitates  a  very  great  increase 
in  pressure,  the  liquid  being  in  general  very  incompressible. 

If  the  same  sequence  of  operations  is  carried  out  at  a  higher 
temperature,  it  is  found  that  condensation  begins  at  a  less  volume 


256  HEAT 

and  higher  pressure,  and  ends  at  a  greater  volume,  that  is,  the 
volume  interval  during  which  the  substance  is  part  vapor  and  part 
liquid  with  a  visible  meniscus  between  becomes  less  as  higher 
temperatures  are  chosen,  until,  when  a  temperature  of  31°  is 
reached,  the  horizontal  part  of  the  isothermal  has  disappeared, 
and  no  separation  into  liquid  and  vapor  can  be  noticed  during 
compression.  The  critical  temperature  is  30.92°G.,  and  it  may 
be  defined  in  another  way  as  that  temperature  above  which  it  is 
impossible  to  liquefy  a  gas  by  pressure  alone,  "to  liquefy"  meaning 
to  cause  a  separation  of  the  two  states. 

The  line  A B  represents  those  conditions  of  pressure  and 
volume  in  which  liquid  and  vapor  can  exist  in  equilibrium  with 
each  other  at  the  temperature  of  21.5°,  and  a  similar  meaning 
attaches  to  the  horizontal  portions  of  the  other  isothermals. 
The  dotted  line  through  the  ends  of  these  horizontal  lines  sur- 
rounds an  area  representing  all  the  physical  conditions  under 
which  the  liquid  and  its  vapor  can  be  in  equilibrium  with  each 
other,  and  the  highest  point  of  this  curve  is  the  critical  point 
whose  coordinates  are  the  critical  volume  and  critical  pressure, 
Vc  and  PC,  corresponding  to  the  critical  temperature  tc. 

318.  The  Saturation  Curve. — That  part  of  the  dotted  curve  to 
the  right  of  the  critical  point  is  called  the  "saturation  cwve" 
and  evidently  represents  all  possible  conditions  of  saturated 
vapor,  and  since  the  diagram  is  drawn  for  unit  mass,  the  ab- 
scissae of  the  points  of  this  curve  are  the  specific  volumes  of  the 
saturated  vapor  at  various  temperatures.  The  left  branch  of  the 
dotted  curve  is  called  the  "liquid  curve"  and  the  abscissae  of  this 
portion  are  the  specific  volumes  of  the  liquid  at  the  same  tempera- 
tures. It  is  very  clear,  then,  that  the  specific  volumes  of  liquid 
and  saturated  vapor  become  equal  at  the  critical  point. 

Above  the  critical  point  the  distinction  between  liquid  and 
vapor  disappears,  and  the  substance  passes  continuously  and 
homogeneously  from  a  rare,  easily  compressible  condition  which 
we  would  call  gaseous,  to  a  dense,  almost  incompressible  condi- 
tion, which  we  would  naturally  call  liquid.  It  is  possible,  by 
properly  varying  the  pressure  volume  and  temperature,  to  pass 
from  any  condition  1  to  any  condition  2  without  crossing  the 
dotted  curve,  that  is,  without  having  the  liquid  distinct  from  the 


CHANGE  OF  STATE 


257 


vapor  at  any  time.     This  property  is  called  the  ''continuity  of 
state." 

It  is  generally  agreed  to  call  a  substance  a  vapor  if  its  condition 
is  represented  by  any  point  below  the  critical  isothermal  and  to 
the  right  of  the  saturation  curve,  and  a  gas  if  represented  by  a 
point  above  the  critical  isothermal,  though  this  distinction  is  not 
important.  The  properties  represented  by  this  set  of  isothermal 
curves  for  C02  are  characteristic  of  all  substances  which  have 
been  studied. 

TABLE  14 
CRITICAL  DATA 


Substance. 

Critical 
temperature  °C. 

Critical 
pressure  (Atmos.). 

Air           

-140 

39 

Alcohol  (ethyl)        

243 

62  7 

130 

115 

Argon        

—  117 

52  9 

Carbon  dioxide              .            . 

30  92 

73 

Chlorine      

146 

93  5 

-268.5 

2.3 

Hydrochloric  acid  

52.3 

86 

Hydrogen             

-234  5 

20 

-146 

33 

Oxvsren 

-118 

50 

Radium  emanation 

104  5 

62  5 

Water           

374 

194  6 

319.  Equations  of  State.  —  Many  attempts  have  been  made  to  derive 
equations  for  the  isothermal  curves  of  Fig.  201,  corresponding  to  the  equa- 
tion PV  =  RT,  which  holds  approximately  for  conditions  far  removed  from 
the  critical,  but  none  have  been  entirely  successful.  One  of  the  most  satis- 
factory of  such  "equations  of  state"  is  that  of  Van  der  Waals: 


in  which  a,  6,  and  R  are  constants  for  a  given  substance.  This  agrees  fairly 
well  with  the  results  of  experiment,  though,  instead  of  the  straight  portion 
A  B  of  the  isothermal,  the  equation  gives  a  continuous  curve  which  cuts  the 
straight  line  in  three  points  as  shown  in  Fig.  202.  The  possibility  of  a  con- 
tinuous passage  such  as  DCBA,  below  the  critical  temperature,  from  the 
vapor  to  the  liquid  condition,  was  suggested  by  James  Thomson  shortly  after 

17 


258 


HEAT 


Andrews'  work;  but,  except  for  portions  of  the  curve  from  A  toward  B 
(under-cooling  a  vapor  free  from  nuclei)  and  from  D  toward  C  (superheating 
a  liquid),  it  has  not  been  realized  experimentally  and  indeed  seems  quite 
unrealizable,  since  it  would  represent  states  in  which  an  increase  in  volume 
would  accompany  an  increase  in  pressure. 

Corresponding  States. — It  was  suggested  by  Van  der  Waals  that  if  the 
pressure,  volume,  and  temperature  were  expressed  in  terms  of  the  critical 
constants,  Pc,  Ve,  tc,  for  each  substance,  as  units,  instead  of  in  atmos- 
pheres, cubic  centimeters  and  degrees  centigrade,  for  example,  the  "equa- 
tion of  state"  would  be  the  same  for  all  substances,  containing  no 
constants  peculiar  to  any  one  material.  The  states  of  all  substances 
would  correspond  when  they  were  represented  by  the  same  "reduced" 
values  of  P,  V,  and  t.  While  this  "theorem  of  corresponding  states"  is  a 
necessary  consequence  of  Van  der  Waals'  equation,  and  may  be  safely  and 
usefully  applied  between  related  substances,  it  is  not  in  general  true. 


OO 

Graph  of 

c^ffX*-**-*' 

for 
/  Gram  ofC02c 

t!js,c 

fin 

S~ 

b 

—  x 

1 

D 

/ 

NJ 

. 

§• 

40 
«<* 

CC 

/ 

\ 

G 

50 

Cubic  Centimeters 
Fio.  202. — Graph  of  Van  der  Waal's  equation. 

320.  Thermodynamic  Surface. — If  the  isothermals  of  Fig.  201  are  placed 
in  their  proper  position  along  the  temperature  axis,  a  smooth  surface  drawn 
through  them  forms  the  thermodynamic  surface  shown  in  Fig.  203,  every 
point  of  which   represents   by  its  coordinates  P,    V,  t,  an  equilibrium 
condition  of  1  gram  of  a  substance.    The  conditions  under  which  the  sub- 
stance may  exist  as  liquid  or  gas  or  as  a  mixture  of  different  states,  are  indi- 
cated  on  the  diagram,   and   the  triple  point  curve  and  the  isothermals 
which  we  have  already  discussed  are  seen  to  be  the  projection  on  the  Pi 
or  Pv  plane  of  lines  on  this  thermodynamic  surface. 

321.  The  Liquefaction  of  Gases. — By  compression  and  cooling 
Faraday  (beginning  in  1823)  liquefied  carbon  dioxide,  sulphur 
dioxide,  chlorine  and  several  other  gases  not  previously  known 


CHANGE  OF  STATE 


259 


in  the  liquid  state.  The  temperatures  he  used  were  evidently 
below  the  critical  temperatures  as  we  now  know  them,  but  the 
problem  was  not  thoroughly  understood  until  the  work  of  An- 
drews made  it  probable  that  extremely  low  temperatures  as  well 
as  high  pressures  would  be  needed  to  liquefy  oxygen,  nitrogen, 
hydrogen  and  air,  which,  as  late  as  1877,  were  called  permanent 
gases.  The  problem  has  been,  then,  one  of  devising  methods  of 
obtaining  extremely  low  temperatures,  and  it  has  been  so  success- 
fully solved  that  all  known  gases  have  now  been  liquefied.  The 
following  methods  are  used  for  obtaining  low  temperatures : 

1.  Chemical  method,  or  the  use  of  freezing  mixtures,  that  is, 
mixtures  of  substances  which  in  dissolving  or  combining  absorb 


Fio.  203. — Thermodynamic  surface,  coordinates  P,  V,  t. 

heat  and  thus  lower  the  temperature.  The  lowest  temperature 
reached  by  this  method,  —  82°C.,  has  been  obtained  by  mixing 
solid  G02  and  liquid  S02.  This  method  is  not  used  in  recent 
liquefying  processes. 

2.  Evaporation  Method. — Fig.  204  illustrates  this  method  as 
applied  to  the  ammonia  refrigeration  process.  A  compressor  ex- 
hausts ammonia  gas  from  above  the  liquid,  compresses  it,  forces 
it  through  tubes  cooled  by  running  water  where  the  heat  of 


260 


HEAT 


vaporization  and  of  compression  is  taken  out  and  it  again 
becomes  liquid,  and  back  through  a  reducing  valve  into  the  evapo- 
rating chamber.  The  evaporating  chamber  is  surrounded  by  the 
material  to  be  cooled  (circulating  brine  in  the  ordinary  refrigerat- 
ing system),  from  which  the  heat  necessary  to  vaporize  the 
ammonia  is  absorbed.  The  ammonia  therefore  absorbs  heat  in 
the  evaporation  chamber  and  loses  heat  in  the  cooling  coils. 

A  series  of  such  circulating  systems,  containing,  for  example,  S02  in  the 
first  and  CO,  in  the  second,  arranged  so  that  the  cooling  of  the  CO,  is  done 
by  the  evaporating  of  the  SOa  is  called  the  cascade  method,  by  which  Pictet 
in  1877  liquefied  oxygen.  The  oxygen  was  compressed  to  several  hundred 
atmospheres  pressure  in  a  tube  surrounded  by  the  evaporating  CO3  and  thus 
cooled  to  —  140°C.  Reference  to  Table  14  shows  that  at  this  temperature 
and  pressure  the  oxygen  would  be  liquid.  Upon  opening  a  cock  the  oxygen 
escaped  in  a  white  stream,  indicating  the  presence  of  the  liquid  or  solid. 
By  adding  lower  steps  to  the  cascade  it  is  possible  to  obtain  very  much 
lower  temperatures,  so  that  oxygen,  nitrogen  and  hydrogen  may  be 
obtained  liquid  at  atmospheric  pressure. 


Jpater 
'^A Outlet 


Fio.  204. — Ammonia  refrigerating  process. 

3.  Regenerative  Unbalanced- expansion  Method. — We  have  al- 
ready considered  (§296)  the  cooling  experienced  by  all  gases 
except  hydrogen  when  forced  through  a  small  opening.  This 
cooling  is  for  air  only  about  0.25°G.  per  atmosphere  decrease 
of  pressure,  but  it  increases  with  decreasing  temperature.  Hence 
if  the  cooled  expanded  gas  is  led  back  around  the  in-flowing 
compressed  gas,  as  in  Fig.  205,  so  as  to  cool  it,  the  temperature  at 
which  expansion  takes  place  will  be  gradually  lowered,  until 
finally  some  of  the  gas  will  liquefy  on  expansion. 

Apparatus  for  applying  this  principle  was  independently  invented  by 
Linde,  Hampson  and  Tripler,  and  this  method  is  now  extensively  used, 


CHANGE  OF  STATE 


261 


commercially  as  well  as  scientifically,  for  liquefying  oxygen,  nitrogen  and 
hydrogen.  The  expenditure  in  the  work  of  compression  of  10  h.p.  for 
one  hour  will,  by  this  process,  produce  from  2  to  4  liters  of  liquid  air,  the 
initial  pressure  being  from  120  to  200  atmospheres.  The  heating  which  is 
observed  with  hydrogen  at  ordinary  temperatures  becomes  zero  at  — 80.5°C., 
as  shown  by  Olzewski,  and  below  that  there  is  a  cooling,  so  that,  by  initially 
cooling  hydrogen  below  —  80.5°C.,  it  can  be  liquefied  by  the  unbalanced 
expansion  process,  as  was  first  done  by  Dewar  in  1898. 

4.  Regenerative  Balanced-expansion  Method. — The  first  step  in 
this  method  is  the  compression  of  the  gas  to  about  40  atmospheres 
pressure,  and  the  partial  cooling  of  it  by  an  "interchange!1" 
analagous  to  the  one  used  in  the  Linde  process.  The  compressed 
and  cooled  gas  is  then  admitted  to  a  cylinder  and  allowed  to 
expand  against  a  piston  thus  doing  external  and  internal  work, 


Dewar 
Liquid  Air^  \^J Vacuum  Flask 

FIG.  205. — Linde's  apparatus  for  liquefying  air. 

and  being  still  further  cooled  to  —  160°G.  or  lower.  The  ex- 
panded gas  is  then  led  around  the  outside  of  a  liquefying  vessel 
containing  air  at  40  atmospheres  pressure,  and  cools  sufficiently 
to  liquefy  some  of  the  air.  The  expanded  gas,  after  absorbing 
heat  from  the  liquefying  vessel,  is  led  back  through  the  inter- 
changer  to  the  compressor. 

The  advantage  of  this  method  over  those  of  the  Linde  type  lies  in  the  greater 
amount  of  external  work  which  the  gas  does,  resulting  in  greater  cooling. 
Moreover,  being  done  against  a  piston,  this  work  can  be  utilized.  By 
combining  three  stages  of  expansion  similar  to  the  above,  Claude  has  pro- 
duced liquid  air  at  the  rate  of  9  liters  per  10  h.p.  per  hour. 


262 


HEAT 


In  1907  helium,  the  last  gas  to  resist  liquefaction,  was  liquefied 
by  Kammerlingh  Onnes  by  the  unbalanced-expansion  method, 
its  boiling-point  under  one  atmosphere  pressure  being  —  268.5°G. 
The  lowest  temperature  attained  by  evaporating  helium  under 
a  pressure  of  1  cm.  of  mercury  was  —  270°G.  or  3°  above  abso- 
lute zero.  These  extreme  temperatures  are  measured  either  by 
a  gas  thermometer  containing  helium  at  reduced  pressure,  or  by 
a  thermo-electric  or  resistance  thermometer. 

TABLE  15 

BOILING-POINTS     OF     DIFFERENT     SUBSTANCES     UNDER     ATMOSPHERIC 

PRESSURE,  AND  TEMPERATURES  OBTAINED  BY  BOILING 

UNDER  REDUCED  PRESSURE 


Substance. 

Boiling-point 
(Atm.  pressure). 

Pressure  (in 
mn£  of  mer- 
cury). 

Boiling-point 
(reduced  pres- 
sure). 

Argon  

-186.2 

300 

-194.2°C. 

Carbon  dioxide 

—  78.2 

2.5 

-130 

Helium  

-268.5 

10 

-270 

Hydrogen                      . 

—  252 

—256 

Neon  

-109 

2.4 

-257.5 

Nitrogen 

—  195  8 

86 

—  210.6 

Oxvcen  . 

—  163 

200 

-194 

Radium  emanation  

-62 

9 

-127 

TRANSFER  OF  HEAT 

322.  Convection,  Conduction  and  Radiation. — Heat  is  trans- 
ferred by  three  very  different  processes. 

Convection  is  the  transport  of  heat  by  moving  matter,  as,  for 
example,  by  the  hot  air  which  can  be  felt  rising  from  a  hot  stove. 

Conduction  is  the  flow  of  heat  through  and  by  means  of  matter 
unaccompanied  by  any  motion  of  the  matter ,  for  example,  the  pas- 
sage of  heat  along  an  iron  bar  one  end  of  which  is  held  in  a  fire. 

Radiation  is  the  passage  of  heat  through  space  without  the 
necessary  presence  of  matter,  for  example,  the  passage  of  heat 
through  the  vacuum  in  the  bulb  of  an  incandescent  lamp. 

323.  Convection. — Convection  occurs  in  liquids  and  gases  and 
is  due  to  the  change  in  density  produced  by  rise  in  temperature. 


TRANSFER  OF  HEAT 


263 


Expansion 
Tank 


A  volume  of  liquid  or  gas  which  varies  in  density  in  different 
parts  is  only  in  stable  equilibrium  when  the  densest  portions  are 
at  the  bottom,  and  there  is  a  regular  decrease  in  density  towards 
the  top.  Since  (with  the  exception  of  water  below  4°G.,  §278), 
liquids  and  gases  expand  on  heating,  thus  diminishing  in  density, 
the  heated  portion  will  rise  and  there  will  be  an  upward  convec- 
tion current  of  hot  substance  and  a  downward  convection  current 
of  cold  substance  to  take  its  place.  If  heat  is  added  at  the  top 
of  an  enclosed  liquid  or  gas,  there  will  be  no  convection,  (except 
with  water  below  4°G.). 

Common  examples  of  convection  by  liquids  are  the  distribution 
of  heat  through  liquids  heated  from  the  bottom,  as  in  the  case  of 
water  in  a  tea  kettle,  and  the  distribution 
of  heat  through  a  house  by  the  hot  water 
system  of  heating,  Fig.  206.  The  water  is 
heated  in  A,  rises  through  B,  is  cooled  in 
the  radiator  and  falls  through  C.  On  a 
large  scale  the  Gulf  Stream,  Japan  current, 
and  other  warm  surface  ocean  currents 
which  start  near  the  equator  are,  in  part 
at  least,  caused  by  convection,  the  return 
being  a  cold  current  flowing  toward  the 
equator  along  the  ocean  bed. 

The  hot-air  furnace  system  of  heating 
houses  is  based  on  convection  by  gases,  the 
hot  air  rising  from  the  furnace  through 
the  pipes  and  registers,  and  the  supply  of 
cold  air  coming  usually  from  outside.  The 
working  of  such  a  system  can  sometimes 
be  improved  by  establishing  a  direct  return 
from  the  coldest  part  of  the  house  to  the  furnace,  thus  com- 
pleting the  indoor  circulation. 

"Natural  ventilation"  is  also  a  convection  process,  an  outlet  being  pro- 
vided at  the  top  of  a  room  for  the  warm  stale  air,  and  an  inlet  at  the  bottom 
for  the  cool  fresh  air.  The  natural  draft  in  chimneys  has  a  similar  cause; 
the  higher  the  chimney  the  larger  is  the  undisturbed  column  of  warm  air 
and  hence  the  greater  the  draft.  The  mixing  of  currents  of  hot  and  cold  air 
usually  cause  a  flickering  or  "boiling"  of  objects  seen  through  them,  because 
light  travels  differently  in  hot  and  cold  air.  This  effect  may  be  seen  by 
looking  across  a  flat  country  in  the  hot  sunshine,  or  over  a  hot  pavement  or 
itove. 


Heater 

Fia.  206.— Transfer  of 
heat  by  convection  of  hot 
water. 


264  HEAT 

The  winds  are  largely  convection  effects,  the  simplest  example 
being  the  "land"  and  "sea"  breezes,  which  ordinarily  blow  from 
the  sea  to  the  land  in  the  morning  and  from  the  land  to  the  sea 
at  night.  These  are  the  return  currents  which  replace  warm  air 
which  rises  from  the  quickly  heated  land  in  the  morning  and 
from  the  warmer,  more  slowly  cooling  sea  at  night. 

CONDUCTION  OF  HEAT 

324.  Characteristics  of  Heat  Conduction. — As  we  have  said, 
conduction  is  the  flow  or  passage  of  heat  energy  through  and  by 
means  of  matter  unaccompanied  by  any  obvious  motion  of 
matter,  as,  for  example,  the  passage  of  heat  through  the  bottom 
of  a  kettle  to  the  water  inside. 

The  direction  in  which  heat  will  flow  between  two  points, 
whether  from  A  to  B  or  B  to  A,  is  found  to  depend  on  the  relative 
temperatures  of  A  and  B,  heat  always  flowing  from  the  point  at 
higher  temperature  to  the  one  at  lower  temperature.  The  greater 
the  difference  of  temperature  beween  two  points,  other  condi- 
tions being  the  same,  the  more  heat  will  flow  per  second,  for 
which  reason  a  kettle  boils  more  quickly  over  a  hot  fire  than  over 
a  low  one.  But,  with  a  given  temperature  difference  between 
the  fire  and  the  water,  boiling  will  take  place  more  quickly  with 
a  thin-bottomed  kettle  than  with  a  thick.  The  temperature 
difference  between  two  points,  A  and  B,  divided  by  the  distance, 

I,  between  them,  or  A""  B)  which  is  the  average  fall  in  tempera- 
ture per  centimeter  between  A  and  B,  is  called  the  temperature 
gradient.  The  above  statements  of  the  dependence  of  conduction 
on  temperature  difference  and  distance  may  be  combined  by 
saying  that  the  amount  of  heat  conducted  per  second  between 
two  points  is  directly  proportional  to  the  temperature  gradient. 
To  pursue  the  same  example,  common  experience  dictates  that 
the  kettle  should  have  a  broad  bottom  in  contact  with  the  stove. 
This  is  an  illustration  of  the  fact  that,  other  conditions  being  the 
same,  the  amount  of  heat  conducted  per  second  is  directly  pro- 
portional to  the  area  through  which  it  can  flow. 

Finally,  the  rate  of  flow  of  heat,  other  conditions  being  the 
same,  depends  greatly  upon  the  material  through  which  it  must 


TRANSFER  OF  HEAT  265 

flow,  substances  being  roughly  divisible  into  "good  conductors/1 
which  permit  under  given  conditions  a  large  flow  of  heat,  and 
which  in  general  are  metallic,  and  "poor  conductors/7  which 
permit  a  small  flow  of  heat  and  are  in  general  non-metallic,  such 
as  wood,  glass,  asbestos,  leather,  linen.  Examples  of  this  differ- 
ence are  very  common.  A  glass  of  hot  water  may  be  handled, 
while  a  metal  cup  containing  the  same  water  will  be  too  hot  to 
touch.  Handles  of  heating  vessels  are  made  of  wood,  or,  if  of 
metal,  are  covered  with  string  or  cloth,  so  that  they  may  be 
touched.  Given  two  bodies,  one  metal  and  one  wood,  at  the 
same  temperature,  below  that  of  the  hand,  the  metal  one  feels 
much  cooler  because  the  heat  it  takes  from  the  hand  quickly 
spreads  through  the  mass,  while,  with  the  poor  conducting  wood, 
the  heat  remains  near  the  surface  of  contact,  which  quickly 
rises  in  temperature.  Thus  the  wood  feels  warmer  because 
(after  the  first  instant)  it  is  warmer,  where  it  is  touched.  The 
rate  of  heat  flow  through  a  body  will  depend  not  only  on  the 
substance  composing  it  but  upon  its  condition  of  subdivision  and 
density.  Thus  saw-dust  conducts  less  readily  than  wood,  and 
the  small  conduction  through  cork  is  partly  due  to  the  reduction 
of  effective  cross-section  by  air  holes.  Also  substances  when 
moist  conduct  better  than  when  dry,  because  water,  which  fills 
the  pores,  is  a  better  conductor  than  air. 

The  characteristic  of  bodies  which  determines  the  rate  of  flow 
of  heat  through  them  is  called  their  thermal  conductivity,  and  the 
numerical  measure  of  this  characteristic,  is  called  the  coefficient 
of  thermal  conductivity  which  will  be  defined  in  the  next  paragraph. 

325.  The  Coefficient  of  Thermal  Conductivity. — In  order  to 
group  together  the  previous  statements  and  obtain  an  exact 
definition  of  the  conductivity  coefficient,  consider  the  passage 
of  heat  from  the  region  (2),  Fig.  207,  at  the  uniform  constant 
temperature  J2,  to  (1)  at  the  temperature  tlt  by  conduction  along 
the  rectangular  bar  of  cross-section  A  ( =  ab)  and  length  I,  no  heat 
being  allowed  to  escape  from  the  sides  of  the  bar.  If  H  is  the 
heat  which  passes  in  a  time  T,  then  from  the  statements  above, 
it  follows  that,  for  a  given  substance, 

(!) 


266 


HEAT 


and,  by  introducing  a  proportionality  factor  K,  properly  chosen 
for  each  substance,  this  may  be  written  as  an  equality, 


(2) 


K  is  the  coefficient  of  thermal  conductivity  of  the  material.  If, 
as  a  special  case,  we  take  A  =  1,  T  =  1, «,-  ^  =  1°,  1  =  1,  then  K  =  #, 
or  K  for  a  given  substance  is  the  heat  flow  per  unit  time  per  unit 
area  with  unit  uniform  temperature-gradient.  In  the  c.g.s.  system, 
the  units  would  of  course  be  the  centimeter,  second,  centigrade- 
degree  and  calorie.  This  statement  defines  K  at  a  definite  mean 
temperature,  (Zi  +  5)°,  but,  since  K  varies  with  the  temperature, 
equation  (2)  will  not  be  true  for  any  great  difference  of  tempera- 
ture, t2  —  tlt  unless  K  stands  for  the  mean  value  between  these 
limits,  and  the  temperature  gradient  is  uniform. 


Fio.  207. — Illustrating  simple  case  of  conduction  of  heat. 

Equation  (2)  is  the  basis  of  most  methods  for  measuring  K, 
but  while  they  are  very  simple  in  principle  they  are  very  difficult 
to  carry  out  accurately. 

If  a  uniform  bar  of  the  substance  has  its  end  faces  maintained 
at  fixed  temperatures,  for  instance  £2  =  100°C.,  and  ^  =  0°G., 
by  contact  with  steam  and  melting  ice  respectively,  its  sides 
being  protected  from  loss  of  heat,  and  if  we  find,  by  measuring 
the  amount  of  ice  melted,  the  heat  which  flows  through  the  bar 
in  a  time  T,  and  also  the  length  and  cross-section  of  the  bar,  K 
may  be  computed  at  once.  This  will,  of  course,  be  the  mean 
value  of  K  between  0°  and  100°G.  Other  calorimetric  methods 
are  usually  used  for  measuring  the  amount  of  heat  flowing  in  a 
given  time,  and  with  poor  conductors  a  slab  or  plate  of  the  sub- 
stance rather  than  a  rod  must  be  used  to  give  a  measurable  rate 
of  flow. 


TRANSFER  OF  HEAT  267 

The  average  temperature  gradient  in  the  earth's  crust  is  about  1°0.  rise 
per  144  ft.  of  descent  near  the  surface,  increasing  to  1°  in  90  ft.  at  depths 
of  a  few  thousand  feet.  This  means  a  continual  loss  of  heat  from  the  interior, 
small  in  amount,  however,  on  account  of  the  low  conductivity  of  rocks  and 
soil.  For  the  same  reason  (low  conductivity)  the  daily  variations  of  surface 
temperature  penetrate  only  about  3  ft.,  the  annual  about  50  ft.  (§238). 

326.  Conduction  in  Liquids. — In  order  to  measure  conduction 
in  liquids,  as  distinct  from  convection,  heat  must,  of  course,  be 
added   at  the  top  (except  with  water  between  0°  and  4°C.), 
since,  if  heated  from  below,  the  warm  expanded  liquid  would 
rise,  while,  if  heated  from  above,  it  will  remain  in  place.     Since 
many  liquids  are  transparent  to  radiation  (§330)  there  is  also 
danger  that  we  may  confuse  radiation  across  the  liquid  with 
conduction  through  it.     The  conductivity  of  liquids  is,  in  general, 
about  that  of  solids  of  low  conductivity,  except  in  the  cases  of 
mercury,  which  is  metallic  and  a  good  conductor,  and  water  and 
some  aqueous  salt  solutions,  which  are  intermediate  between 
the  metallic  and  the  non-metallic  solid  conductors. 

327.  Conduction  in  Gases. — The  masking  of  conduction  by 
convection  and  radiation  is  even  more  likely  to  occur  in  gases, 
because  of  their  greater  mobility,  greater  transparency,  and 
lower  real  conductivity.    The  conductivity  of  hydrogen  and 
helium  is  much  greater  than  that  of  other  gases.     This  follows 
naturally  from  the  kinetic  theory,  since  they  have  the  smallest 
molecular  weights,  therefore  the  highest  molecular  velocities  at  a 
given  temperature,  and  therefore  hand  on  kinetic  energy  from 
molecule  to  molecule  with  the  greatest  rapidity.     The  conduc- 
tivity of  gases  is,  through  wide  ranges,  independent  of  the  pressure 
as  theory  also  indicates  should  be  the  case.     On  account  of  their 
extremely  low  conductivity  air  layers  enclosed  in  or  between 
solids,  such  as  air  spaces  in  house  walls  and  in  the  walls  of  refrig- 
erators, in  pores  in  cloth  and  in  fur  or  feathers,  are  chiefly  respon- 
sible for  the  low  conductivity  found  in  these  cases. 

328.  Conductivity  of  Alloys  and  Crystals. — The  conductivity  of  copper  is 
increased  by  compression,  that  of  steel  diminished  by  hardening.     The  con- 
ductivity of  alloys  is  not,  in  general,  simply  proportional  to  the  relative 
amounts  of  the  pure  metals  forming  the  alloy,  but  may  have  decided  mini- 
mum values  in  case  compounds  are  formed.     In  non-isotropic  solids,  such  as 
wood  and  crystals,  the  conductivity  depends  upon  the  direction  of  flow, 
being  in  the  case  of  wood  two  or  three  times  as  great  along  the  fiber  as  at 


268  HEAT 

right  angles  to  it.  In  crystals  the  axes  of  symmetry  for  heat  conduction 
coincide  with  the  crystalline  axes,  and  the  conductivity  is  different  in 
different  directions,  as  may  be  very  prettily  shown  by  means  of  a  thin  plate 
of  crystal  coated  with  wax  on  one  side,  and  having  a  wire  passing  normally 
through  the  center.  If  the  wire  is  carefully  heated  the  wax  will  gradually 
melt  and  the  limit  of  melting  will  be,  in  general,  an  ellipse  and  not  a  circle, 
as  it  would  be  with  an  isotropic  plate.  The  marked  decrease  in  effective 
conductivity  resulting  from  breaking  up  a  solid  has  been  referred  to,  but 
this,  as  well  as  the  effect  of  compression  on  a  substance  like  felt  or  cotton,  is 
not  a  change  in  the  property  of  the  substance  itself,  but  merely  a  change  in 
the  amount  of  poorly  conducting  material  (air)  mixed  with  it. 

TABLE  16 
THERMAL  CONDUCTIVITIES 

(c.g.s.  units.) 
Substance.  Conductivity.     Substance.  Conductivity. 

Aluminum 0.504  Lead 0.083 

Brass 0.260  Nickel 0.142 

Air 0.00005        Oak 0.0006 

Concrete 0.0022          Platinum 0. 166 

Copper 0.918  Porcelain  (Berlin) 0.0025 

Cork 0.00013         Quartz,  ||-axis 0.030 

Cotton  wool 0.00004         Quartz,  J_-axis 0.016 

Earth's  crust 0.004  Sawdust 0.00012 

Flannel 0.00023         Silk 0.00022 

Glass 0.0024          Silver 0.974 

Gold 0.700  Tin 0.155 

Ice 0.005  Water 0.0014 

Iron 0.144  Zinc 0.265 

329.  The  Nature  of  Conduction. — Since  we  have  agreed  that 
heat  energy  is  in  large  part  kinetic  energy  of  motion  of  molecules, 
atoms,  and  electrons,  it  is  natural  to  think  of  this  motion  (heat) 
as  spreading  through  a  substance  by  collision  of  these  particles 
with  each  other.  Heat  added  to  one  side  of  a  body  will  increase 
the  average  energy  of  motion  of  the  molecules  on  that  side,  and 
will  be  gradually  handed  on  by  impact  to  the  slower  moving 
ones  and  so  will  spread  through  the  mass,  much  as  a  disturbance 
originating  at  one  point  would  spread  through  a  closely  packed 
crowd  of  people  by  repeated  pushing  and  jostling. 

While  this  has  been  the  common  idea  of  the  nature  of  the  process,  J.  J. 
Thomson  and  others  have  recently  attempted  to  account  for  the  matter  in 
an  entirely  different  way,  namely,  by  the  convection  of  "free"  electron!  as 


TRANSFER  OF  HEAT  269 

defined  in  §262.  According  to  this  hypothesis  the  addition  of  heat  to  one 
part  of  a  substance  increases  the  kinetic  energy  of  the  free  electrons  in  this 
region,  and  there  results,  not  only  the  transfer  of  energy  by  impact,  but  the 
actual  diffusion  of  fast  moving  electrons  from  the  hot  to  the  cold  region. 
The  motion  of  these  electrons,  according  to  this  hypothesis,  also  constitutes 
an  electric  current,  so  that  this  new  explanation  of  heat  conductivity  would 
very  easily  account  for  the  remarkable  observed  fact  that  those  substances 
which  conduct  heat  readily,  such  as  metals,  also  conduct  electricity  readily, 
the  electrical  and  thermal  conductivities  being  in  a  fairly  constant  ratio  for 
most  metals  at  ordinary  temperatures. 

RADIATION 

330.  Radiant  Energy. — Radiation  is  the  process  by  which 
energy  is  transmitted  through  space  without  the  necessary  pres- 
ence of  matter.  While  being  transmitted  in  this  way  energy  is 
called  radiant  energy,  and  is  not  heat,  since  the  latter  is  energy  in 
a  particular  relation  to  matter.  That  energy  may  pass  through 
matter  and  still  not  be  heat  may  be  shown  by  allowing  the  sun's 
rays  to  pass  through  glass  and  fall  upon  a  blackened  thermom- 
eter, which  may  be  very  decidedly  heated,  though  the  glass 
remains  cool.  Radiation  differs  most  strikingly  from  convection 
and  conduction  in  speed.  Time  a  convection  current  (by  means 
of  smoke  or  dust)  and  the  velocity  will  usually  not  be  many  feet 
per  second;  thrust  one  end  of  a  silver  rod  into  hot  water  and  it 
will  be  several  seconds  before  a  noticeable  effect  can  be  felt  a  few 
centimeters  above  the  surface;  but  an  opaque  screen  for  cutting 
off  radiation  produces  a  practically  instantaneous  effect  even  at  a 
great  distance. 

The  early  idea  regarding  the  nature  of  radiation  was  that 
it  was  a  streaming  of  fine  particles — that  is,  a  convection.  It  is 
now  known  to  be  a  wave  disturbance,  such  as  has  been  discussed 
in  Wave  Motion,  analogous  to  the  waves  which  travel  over  a 
water  surface.  The  "disturbance"  of  which  the  water  waves 
consists  is  an  up-and-down  motion  of  the  water  particles,  and 
this  disturbance  travels  forward  while  the  water  moves  up  and 
down.  The  "disturbance"  in  a  radiation  wave  is  a  transverse 
(§238)  electric  (and  magnetic)  force  (§543),  which  changes  in 
direction  and  amount  as  a  wave  passes  a  point  (just  as  the  motion 
of  the  water  particles  changes  from  up  to  down),  and  which  would 
move  a  compass  needle  if  we  could  make  one  small  enough  for 


270 


HEAT 


it  to  act  on.  The  characteristics  of  a  radiation  wave  are  the 
period,  the  wave-length,  and  the  magnitude  of  the  electric  force 
which  a  wave  produces  as  it  passes,  or  the  amplitude,  correspond- 
ing to  the  height  of  a  "crest"  in  a  water  wave,  which  determines 
how  strong  the  wave  is,  how  much  energy  it  represents  (§259), 
and  is  quite  independent  of  the  wave-length.  A  strong  wave 
may  be  long  or  short,  a  long  wave  weak  or  strong. 

331.  Light  and  Radiation. — Radiation  travels  with  the  same 
speed  as  light  (§641),  and  like  light  it  can  be  reflected  by  mirrors 
and  refracted  by  lenses  and  prisms.     These  and  other  facts  prove 
conclusively  that  radiation  waves  are  of  exactly  the  same  nature 
as  light  waves,  in  fact  that  light  consists  simply  of  those  radiation 
waves  whose  lengths  lie  between  0.0004  and  0.00076  millimeters 
and  which  affect  the  eye.    These  waves  lie  near  one  end  of  the 
entire  known  range  of  radiation  wave-lengths,  which  is  from 

0.0001  mm.  to  0.3480  mm.  and  is  called 
the  radiation  spectrum.  Of  this  the  visi- 
ble spectrum  evidently  forms  but  a  very 
small  part.  It  is  sometimes  convenient 
to  represent  the  spectrum  by  points 
along  the  z-axis  whose  abscissse  are  pro- 
portional to  the  wave-lengths.  That 
part  of  the  spectrum  called  the  "infra- 
red," of  longer  wave-length  than  the 
visible,  contains  usually  the  waves  of 
greater  energy,  the  most  important  for 
the  radiation  of  heat,  and  these  waves 
were  formerly  called  "radiant  heat." 

Radiation  waves  can  travel  through  free  space,  their  transmis- 
sion being  one  of  the  fundamental  properties  of  space  as  we 
know  it. 

332.  Law  of  Exchanges. — If  two  bodies  at  different  tempera- 
tures, for  example  the  copper  balls  A  and  B,  Fig  208,.  are  put  in  a 
vacuum  and  not  in  contact,  equality  of  temperature  will  be  es- 
tablished by  radiation,  the  hotter  body  A  on  the  whole  radiating 
heat  to  the  colder  B.     If,  without  changing  the  temperature  or 
condition  of  B,  A  is  cooled  till  it  is  the  colder  of  the  two,  the  net 
exchange  of  heat  by  radiation  will  now  be  from  B  to  A.     Since 
we  have  not  altered  B  in  any  way,  we  conclude  that  B  was 


Fio.  208. — Radiating  bodies 
in  a  vacuum. 


TRANSFER  OF  HEAT 


271 


radiating  to  A  in  the  first  case  also,  but  that  A  was  radiating 


then   always   a  reciprocal  process, 


more  to  B.  Radiation  is 
or  one  of  exchange.  This 
is  the  celebrated  Prevost 
law  of  exchange,  according 
to  which  radiation  equilib- 
rium is  the  result  of  equal 
streams  of  radiant  energy 
in  opposite  directions,  and 
does  not  indicate  the  ces- 
sation of  radiation. 

333.  Measurement  of 
Radiation. — In  order  to 
measure  radiation  it  is 
converted  into  heat  by 
absorption  in  matter,  the  

heat  being  then  measured  FIG.  209.— Thermopile  for  measuring  radiant  energy 

by  the  temperature  change 

which  it  produces.  To  make  this  process  delicate,  recourse  is 
had  to  the  thermo-electric  or  resistance  methods  for  measuring 
temperature  described  in  §§268,  269. 

The  thermopile.  Figs.  209, 210,  consists  of  one  or  more 
"junctions"  of  different  metals,  iron  and  constan- 
tan,  or  better  two  al  oys  of  bismuth-antimony  and 
antimony-cadmium  arranged  as  shown  in  Fig.  210, 
so  that  one  set  of  similar  junctions  can  be  exposed  to 
radiation  while  the  other  set  is  protected.  To  increase 
the  amount  of  radiant  energy  intercepted,  the  exposed 
junctions  should  be  covered  with  light  blackened  silver 
or  copper  disks,  and  similar  disks  should  be  put  on 
the  other  junctions.  If  the  final  elements  are  con- 
nected by  wires  to  a  very  delicate  galvanometer,  very 
slight  changes  in  temperature  of  one  set  of  junctions, 

of  the  order  of  jTooolxx)0^''  or  ^e88'  w^  produce  a 
readable  deflection,  and  will  correspond  to  a  very  weak 
stream  of  radiant  energy  falling  on  the  exposed  junc- 
tions, such  as  for  example,  the  radiation  from  a  single 
candle  at  a  distance  of  50  meters.  To  be  quick- 
acting  and  sensitive  the  mass  of  the  junctions  should  be  small. 

The  bolometer,  Fig.  211,  is  an  even  more  sensitive  instrument.  It  con- 
sists essentially  of  two  similar  strips  of  very  thin  (0.001  mm.)  blackened 
platinum  mounted  side  by  side,  having  exactly  the  same  resistance,  and 


Detail  of  Junctions 
Fio.  210.— Detail 
showing  arrangement 
of  the  exposed  (inner) 
and  protected  (outer) 
junctions  of  a  thermo- 
pile. 


272 


HEAT 


arranged  in  a  Wheatstone  bridge  (§456),  BO  that  any  unequal  changes  in 
resistance  of  the  strips  can  be  very  sensitively  measured.  If  one  strip  is 
exposed  to  radiation,  its  temperature  and  hence  its  resistance  will  change. 

334.  Emission,  Absorption  and  Reflection. — Emission  is  the 
starting  of  radiation  waves.  The  conversion  of  the  energy  of  a 
wave  into  heat  by  passage  through  matter  is  called  absorption.  A 
substance  is  opaque  to  radiation  when  it  will  not  allow  the  radia- 
tion to  pass  through,  as,  for  example,  wood  and  metals  are  opaque 
to  light.  Absorption  by  a  very  thin  surface  layer  of  a  strongly 
absorbing  substance  is  called  surface  absorption.  Such  a  surface, 
if  polished,  will  also,  in  general,  reflect  very  well. 


Fio.  211. — Bolometer  for  measuring  radiant  energy,  and  Wheatatone's  bridge  for  measuring 
change  of  its  resistance. 


The  absorbing  power  of  a  surface  is  the  ratio  of  the  radiant 
energy  absorbed  by  the  surface  to  the  amount  incident  upon  it. 
The  absorbing  power  of  a  given  surface  is,  in  general,  different 
or  different  wave-lengths  of  radiation.  Let  A^  be  the  absorbing 
power  for  the  wave-length  ^  and  A  the  total  absorbing  power  for 
all  wave-lengths.  The  values  of  A  and  Ax  are  practically  inde- 
pendent of  temperature.  The  emissivity  of  a  surface  is  the  total 
radiant  energy,  in  ergs,  which  the  surface  sends  out  per  square 
centimeter  per  second,  this  radiation  being  caused  by  the  heat  of 
the  surface.  The  hotter  a  body  the  more  it  radiates,  that  is, 
emissivity  increases  with  temperature.  We  shall  denote  the 
total  emissivity  for  all  wave-lengths  by  E,  and  the  emissivity  for 
the  wave-length  A,  or  partial  emissivity,  by  E^ 


TRANSFER  OF  HEAT  273 

The  reflecting  power  of  a  surface  is  the  ratio  of  the  radiant 
energy  reflected  from  the  surface  to  the  energy  incident  upon  it. 
The  reflecting  power  of  a  surface  is  different  for  different  substances 
and  for  different  wave-lengths  of  radiation.  Thus  a  polished 
silver  surface  will  reflect  about  82  per  cent,  of  all  blue  light 
falling  upon  it,  about  92  per  cent,  of  incident  yellow  light,  and 
about  98  per  cent,  of  energy  in  the  form  of  long  infra-red  waves, 
while  a  polished  iron  surface  reflects  about  57  per  cent,  of  yellow 
light  and  from  78  to  97  per  cent,  of  the  energy  of  infra-red  waves. 

That  there  is  a  close  connection  between  the  absorbing  power 
and  emissivity  of  a  surface  can  be  shown,  for  example,  by  heating 
a  bit  of  white  china  with  blue  markings,  which  look  dark  against 
the  light  china  at  ordinary  temperatures  because  they  absorb 
more  light,  but  look  bright  against  the  china  at  high  temperatures, 
showing  that  they  emit  more  light.  Similarly  black  ink  marks 
on  platinum  look  bright  when  heated.  In  general  good  absorbers 
are  good  radiators.  That  this  must  be  so  follows  from  a  con- 
sideration of  a  body  B  suspended  inside  an  exhausted  opaque 
vessel  C.  Experience  shows  that  B  and  C  will  come  to  the 
same  temperature  by  interchange  of  radiation,  and  when  equi- 
librium is  reached  B  must  absorb  per  second  as  much  as  it 
radiates.  Hence  if  B  is  a  good  absorber  it  must  be  a  good 
radiator,  and  vice  versa. 

335.  Kirchhoff's  Law.  —  The  exact  relation  between  absorbing  power  and 
emissivity.  deduced  theoretically  by  Kirchhoff  and  called  after  him  Kirch- 
hoff's  Law,  is  that  the  ratio  of  the  emissivity  to  the  absorbing  power  is  the  same 
for  all  surfaces  at  any  one  temperature,  or 


and  similarly,  as  regards  any  particular  wave-length, 


where  E  and  E;  are  constants  independent  of  the  substances. 

336.  A  Perfect  Absorber  and  Perfect  Radiator.  —  If  we  could  have  a 
surface  which  absorbed  all  the  radiation  ailing  upon  it,  called  a  perfect 
absorber  or  black  body,  then  for  this  surface 

4—  4;t  —  1,  and  consequently  E**E  and  J?j  —  Ej. 

In  other  words,  the  constants  E  and  Ej  are  the  total  and  partial  emissivi- 
ties  of  a  black  body.     Since  A  and  A^  can  never  be  greater  than  1,  it 
follows  that  a  black  body  has  the  greatest  possfole  total  and  partial  emissivity, 
18 


274  HEAT 

at  any  temperature,  and  it  is,  therefore,  also  called  a  perfect  radiator.  A 
hollow  opaque  body  having  a  small  opening  in  the  walls  is  a  very  close 
practical  approximation  to  a  black  body,  because  radiation  entering  through 
the  opening  is  partially  reflected  and  re-reflected  inside  and  thus  eventually 
almost  all  absorbed.  Also  a  sharp  conical  hollow  or  wedge-shaped  cleft 
with  straight  opaque  polished  sides,  no  matter  of  what  they  are  made, 
absorbs  all  radiation  entering  it.  Conversely,  if  the  walls  of  the  enclosure, 
or  cone,  or  cleft  are  uniformly  heated,  the  radiation  which  leaves  the  opening 
will  be  that  of  a  perfect  radiator  at  the  temperature  of  the  walls,  since,  as 
we  concluded  in  §335,  it  must  be  independent  of  the  nature  of  the  enclosure. 
These  are  all  practicable  ways  of  realizing  a  perfect  absorber  and  perfect 
radiator. 

337.  Total  Radiation  and  Temperature. — The  radiation  of  all 
bodies  increases  with  the  temperature,  but  the  laws  governing 
this  increase  are  not  as  yet  known  except  for  a  perfect  radiator,  for 
which  Boltzmann  deduced  in  1883  the  law  previously  suggested 
by  Stefan,  that 

E=sT*,  (1) 

T  being  the  absolute  temperature  of  the  surface  and  s  a  constant 
which  later  work  has  shown  to  be  approximately  5.6X10"* 
ergs  per  square  centimeter  per  second.  According  to  this  law  the 
radiation  from  one  square  centimeter  of  black  body  surface  at 
400°  absolute  (127°C.)  would  heat  one  gram  of  water  1.5°G.  per 
minute.  If  one  black  body  surface  at  temperature  T  is  radiating 
to  another  surrounding  it  at  temperature  T19  then  the  net  or 
differential  radiating  power  will  be,  from  the  law  of  exchanges, 

E=s(T'-T*) 

While  this  law  can  be  deduced  only  for  a  black  body,  it  is  found 
to  hold  approximately  for  other  surfaces.  Rewriting  it  in  the 
form 

E  =  s(T-  7\) (T3  +  T27\  +  TTS  +  2V), 

it  is  evident  that,  if  Tl  does  not  differ  much  from  T,  we  have 
approximately 

E  =  4sT*(T-  T,)  =K(T-  T,)  (for  T  constant)  (2) 

A  similar  relation,  known  as  Newton1  s  law  of  cooling,  is  found 
to  hold  for  the  loss  of  heat  by  combined  radiation  and  convection, 
and  was  enunciated  by  Newton  as  follows: 

The  heat  lost  by  radiation  and  convection  by  one  body  to  another 
surrounding  it  is  proportional  to  the  temperature  difference  between 


TRANSFER  OF  HEAT 


276 


the  two.     This  is  a  convenient  relation  to  use  and  is  quite  accurate 
for  small  temperature  differences. 

338.  Distribution  of  Energy  in  the  Spectrum. — As  the  tempera- 
ture of  any  radiating  surface  is  raised,  the  energy  emitted  in 
every  wave-length  increases  also,  but  not  in  equal  proportion. 
It  is  a  matter  of  common  experience  that  the  light  emitted  from  a 


.007 


.002  .003  .004 

Wave  Length  in  m.m- 


.005 


.006 


Fio.  212. — Curves'showing  distribution  of  energy  in  the  spectrum  of  a  perfect  radiator  at 

various  temperatures. 

hot  radiating  surface  changes  in  color  as  the  temperature  of  the 
surface  is  raised,  changing  from  red  to  yellowish,  then  to  white 
and  finally  having  a  blue-white  color  at  extremely  high 
temperatures. 

If,  for  a  given  surface,  we  plot  the  values  of  E^  as  ordinates,  and 
the  corresponding  values  of  X  as  abscissae,  for  any  one  temperature, 
we  obtain  what  is  called  the  energy  curve  for  this  temperature. 


276  HEAT 

For  example,  the  energy  curves  of  Fig.  212  show  clearly  the  dis- 
tribution of  energy  in  the  spectrum  of  a  perfect  radiator  at  several 
temperatures.  Such  curves  have  a  general  similarity  for  all 
surfaces,  the  emission  being  weak  for  short  wave-lengths,  rising 
to  a  maximum,  and  diminishing  again  for  long  wave-lengths. 
As  the  temperature  of  the  radiating  surface  is  increased,  all  the 
ordinates  of  the  curve  increase,  and  the  maximum  shifts  toward 
the  short  wave-lengths.  This  shift  of  the  energy  curve,  resulting 
in  an  increasing  proportion  of  blue  in  the  emitted  light,  accounts 
for  the  change  in  color  of  an  incandescent  body,  which  was  just 
referred  to.  For  a  perfect  radiator  at  100°C.  the  maximum  of 
the  energy  curve  lies  at  a  wave-length  of  about  0.008  mm.,  while 
for  carbon  at  the  temperature  of  the  arc  it  has  shifted  to 
the  edge  of  the  visible  spectrum,  and  for  the  sun  it  is  in  the 
yellow. 

The  emission  (Ex)  for  any  wave-length  for  a  black  body  is 
given  with  very  great  accuracy  by  the  expression 

log^=^+^  (3) 

Where  K^  and  K2  are  constants  and  T  is  the  absolute  tempera- 
ture. For  some  other  radiating  surfaces  E^  has  been  found  to 
follow  quite  closely  the  same  law,  though  with  different  constants. 
339.  Radiation  Pyrometry. — By  this  is  meant  the  measure- 
ment of  high  temperatures  by  observing  the  variation  with 
temperature  of  either  total  emissivity  or  partial  emissivity.  In 
the  former  case  radiation  of  all  wave-lengths  from  the  surface 
whose  temperature  is  to  be  me'asured  is  allowed  to  fall  on  a 
thermopile  of  some  sort  and  the  resulting  deflection  of  a  volt- 
meter or  galvanometer  is  noted.  By  observations  on  a  surface 
at  known  temperatures  the  instrument  can  be  calibrated  so  as 
to  indicate  temperatures  directly.  Instruments  of  this  sort  are 
the  Fe"ry  or  Thwing  total  radiation  pyrometers.  If  the  instru- 
ment is  calibrated  by  using  a  perfect  radiator,  and  used  on  another 
surface,  it  will  indicate,  not  the  true  temperature  of  this  surface, 
but  the  temperature  of  a  perfect  radiator,  which  would  radiate 
with  the  same  total  intensity  as  this  surface.  This  is  called  a 
black  body  temperature  of  the  surface,  and  will  usually  be  lower  (it 
cannot  be  higher)  than  the  true  temperature. 
-  Optical  pyrometry  makes  use  of  the  partial  emissivity.  The 


THE  CONSERVATION  OF  ENERGY 


277 


method  consists  in  comparing  the  radiation  of  a  given  wave- 
length (usually  red)  from  the  surface  whose  temperature  is 
desired  and  from  a  comparison  source,  usually  a  small  incan- 
descent lamp.  In  using  the  instrument  the  electric  current  is 
measured  which  is  required  to  heat  the  comparison  lamp  so 
that  it  disappears  when  viewed  against  the  hot  body.  The 
instrument  is  calibrated  by  observations  on  a  black  body  at 
known  temperatures,  and  the  radiation  laws  given  by  the  equa- 
tion on  p.  276  may  be  used  to  extend  the  scale  beyond  the  region 
of  possible  comparison.  In  this  way  measurements  have  been 
made  up  to  3600°C.  If  used  on  a  surface  other  than  a  perfect 
radiator,  it  will  give  a  "black  body  temperature,"  less  than  the 


\ 

| v/WWVWV  \ 


Fia.  213. — Optical  pyrometer  for  high  temperature  measurements.     Ammeter  for  measuring 
current  in  comparison  lamp. 

true  temperature.  Examples  of  this  type  of  instrument  are  the 
Holborn,  Morse,  and  Wanner  optical  pyrometers,  the  first  named 
being  shown  in  Fig.  213.  Radiation  pyrometry  at  present  is 
the  only  satisfactory  method  available  above  about  1750°G. 

THE  CONSERVATION  OF  ENERGY 

340.  The  Transformation  of  Mechanical  Energy  into  Heat. — 
Since  heat  is  energy  and  can  be  produced  by  the  transformation 
of  mechanical  energy,  it  is  of  great  importance  to  determine  just 
how  much  mechanical  energy  is  equal  to  unit  quantity  of  heat. 
In  the  c.g.s.  system,  the  mechanical  equivalent  of  heat  is  the 
number  of  ergs  equivalent  to  (i.e.,  which  will  produce)  one  calorie. 
The  symbol  for  it  is  /. 

The  first  careful  determination  of  this  important  quantity 
was  by  Joule  of  Manchester  in  1843,  before  the  caloric  theory 
was  finally  overthrown.  Rowland  in  1878  carried  out  one  of 
the  most  reliable  determinations  of  J  which  have  been  made,  the 
method  being  an  improvement  of  one  used  by  Joule  many  years 


278 


HEAT 


before.  A  calorimeter  (Fig.  214)  contains  water  and  one 
fixed  and  one  movable  set  of  paddles,  the  latter  driven  by  a 
shaft  through  the  bottom  of  the  calorimeter,  and  the  former  so 
arranged  that  the  water  can  not  rotate  as  a  mass,  but  will  be 
violently  churned.  The  paddles  are  driven  at  a  steady  speed 
by  a  steam  engine,  and  the  calorimeter  prevented  from  rotating 
by  the  couple  applied  through  two  cords  which  pass  tangentially 
from  a  carefully  turned  rim  of  radius  R,  and  after  passing  over 
frictionless  pulleys  carry  two  weights  of  M  grams  each.  The 
resisting  couple  experienced  by  the  paddles  in  their  motion  through 
the  water  must  be  equal  in  magnitude  to  the  couple  which  the 


Belt 
Fio.  214.  —  Apparatus  for  measuring  the  mechanical  equivalent  of  heat. 

water  exerts  tending  to  rotate  the  calorimeter.  The  resisting 
couple  increases  as  the  speed  of  rotation  is  increased.  If  for  a 
given  speed  the  weights  M  are  so  adjusted  that  the  calorimeter 
does  not  rotate,  and  if  we  let  L  represent  the  moment  of  the 
resisting  couple,  then 

L=2RMg. 
From  this  the  work  done  by  the  paddles  can  be  calculated. 

Work  in  one  revolution 

Work  in  N  revolutions 


If,  after  the  N  revolutions,  the  temperature  of  the  calorimeter  has 
risen  (tt—  tj0,  and  if  m  and  m'  are  the  mass  of  water  contained  in, 


THE  CONSERVATION  OF  ENERGY  279 

and  the  water  equivalent  of  the  calorimeter  respectively,  then 
(m  +  m')  (^-y  =the  heat  added  to  and  retained  by  the  water. 
To  this  must  be  added  the  heat  lost  by  convection  and  radiation, 
which  we  shall  denote  by  H.  There  is  no  correction  to  be  ap- 
plied to  the  expression  for  the  work,  for  only  the  work  done  against 
the  frictional  forces  in  the  calorimeter  is  measured.  Hence 
(InNRMg)  ergs  are  equivalent  to  (ra  +  ra')  (£2  —  tj  +H  calories 

InNRMg 
^-fr  +  mOO.-O+ 

Allowance  must  also  be  made  for  variation  in  the  specific  heat  of 
water. 

Other  methods  have  been  used  in  which  the  friction  occurred 
between  metal  surfaces,  the  heat  being  absorbed  from  them  by 
the  water  of  the  calorimeter  or  by  a  stream  of  water  flowing  past 
them  (see  §289).  The  latter  arrangement  is  identical  with  one 
of  the  standard  forms  of  absorption  dynamometers  used  in  en- 
gineering practice  for  obtaining  the  power  developed  by  an  en- 
gine or  motor.  Since  the  mechanical  value  of  electrical  work  can 
be  very  accurately  determined  (§458),  it  is  possible  to  determine 
/  indirectly  by  converting  electrical  energy  into  heat,  as  has 
been  done  by  Callendar  and  others.  This  method  is  simpler  than 
the  direct  one.  The  average  of  the  four  best  determinations  is 


Cal18 

which  is  practically  Rowland's  value  and  is  probably  correct  to 
-fa  per  cent.  The  numerical  value  of  J  depends,  of  course,  on  the 
units,  the  following  being  also  used 

,7  =  427   kilogram-meters   per  large   calorie. 
J  =  778  foot-pounds  per  B.  T.  U. 

341.  The  Law  of  Conservation  of  Energy. — We  have  already 
become  familiar  in  Mechanics  with  the  transformation  of  kinetic 
energy  (Jmv2)  into  potential  energy  when  work  is  done  against 
mechanical  forces,  and  we  have  given  reasons  for  believing 
that  heat  is  a  special  form  of  kinetic  and  potential  energy. 


280  HEAT 

Later  we  shall  have  to  deal  with  electric  and  magnetic  forces  and 
work  done  against  them,  giving  us  the  idea  of  electric  and  mag- 
netic kinetic  and  potential  energy,  while  chemistry  deals  with 
chemical  potential  energy,  though  this  may  ultimately  be  found 
to  be  electrical  in  nature. 

After  the  growth  of  the  idea  that  heat  is  energy,  and  Joule's 
early  (1843)  determination  of  J,  Helmholtz,  in  1847,  formulated 
the  idea  that  not  only  heat  and  mechanical  energy,  but  all 
forms  of  energy  are  equivalent,  and  that  a  given  amount  of  one 
form  cannot  be  made  to  disappear  without  an  equal  amount  ap- 
pearing in  some  of  the  other  forms.  For  example,  when  the 
potential  energy  of  a  wound  clock  spring  disappears,  heat,  caused 
by  work  done  against  frictional  forces,  appears  in  the  clock, 
while  energy  of  sound  waves  and  kinetic  energy  of  motion  of 
parts  of  the  clock  are  also  produced.  Again,  the  heat  energy  of 
steam  may  be  transformed  into  mechanical  energy  by  a  steam 
engine  and  given  to  a  dynamo,  which  does  work  against  electric 
and  magnetic  forces,  producing  some  heat  but  largely  electric 
potential  energy,  which  in  turn  is  changed,  by  the  flow  of  an 
electric  current,  partly  into  heat  in  the  wire,  but  largely  into 
mechanical  work  by  a  motor,  or  into  light  and  heat  by  an  electric 
lamp.  This  idea  of  equivalence  may  be  expressed  in  many 
ways,  such  as, 

Energy  is  indestructible. 

The  total  amount  of  energy  in  the  universe  is  constant. 

The  energy  required  to  change  a  system  of  bodies  from  one  state 
(including  of  course  its  electric  and  magnetic  condition)  to  another 
state  is  independent  of  the  particular  intermediate  states  through 
which  it  passes. 

These  are  all  statements  of  the  Law  of  Conservation  of  Energy , 
of  which  the  last  is  perhaps  the  best,  because  we  cannot  deal  with 
the  universe,  nor  can  we  measure  the  total  amount  of  energy 
present  in  any  body.  The  fundamental  idea  is  that  all  processes, 
such  as  the  change  of  the  energy  of  steam  into  mechanical  energy 
and  light  above  mentioned,  consist  in  drawing  a  stream  of  energy 
from  some  source  and  then  dividing  and  diverting  that  stream 
into  various  channels  such  as  heat,  mechanical  work,  light,  etc. 
Common  experience  shows  us  that  it  is  always  very  easy  to  con- 
vert any  other  form  of  energy  into  heat.  Whenever  a  bell  is 


THERMODYNAMICS  281 

rung  by  a  battery,  or  a  pump  operated  by  a  wind  mill,  some  of 
the  energy  of  the  battery  or  the  wind  is  changed  into  heat. 

Like  all  the  greatest  fundamental  physical  laws,  the  law  of 
conservation  of  energy  is  not  capable  of  direct  proof,  but  is  a 
hypothesis  consistent  with  all  known  facts,  which  is  to  be  accepted 
until  some  phenomena  are  discovered  with  which  it  is  inconsis- 
tent. It  is  of  the  widest  possible  application  and  is  the  chief  basis 
of  all  physical,  astronomical  and  chemical  reasoning,  as  well  as  of 
engineering  practice.  It  leads  us  to  doubt  at  once  all  "perpetual 
motion"  devices  which  purport  to  obtain  mechanical  work  from 
nothing. 

THERMODYNAMICS 

342.  First  Law  of  Thermodynamics. — Thermodynamics  is  the 
analysis  and  discussion  of  the  problems  of  converting  heat  into 
other  forms  of  energy,  and  other  forms  of  energy  into  heat,  and 
consists  in  the  deduction  of  consequences  from  two  very  general 
principles,  the  first  one  being  the  law  of  conservation  of  energy 
(§341). 

Considering  a  body  or  a  system  or  group  of  bodies  as  distinct 
from  its  surroundings,  we  have  already  (§262)  defined  the  term 
"internal  energy/'  for  which  we  shall  use  the  symbol  U,  as 
the  entire  energy  which  the  system  contains.  As  was  pointed 
out  earlier,  we  have  no  knowledge  of  the  value  of  U  in  any  case, 
but  we  can  study  the  changes  in  U.  If  the  reactions  between  the 
system  and  outside  bodies  are  such  as  to  permit  the  passage  of 
heat  to  or  from  the  system,  and  the  doing  of  work  on  or  by  the 
system,  then  it  follows  from  the  law  of  conservation  of  energy, 
that  for  any  change  in  the  system 

the  increase  in  \       *»  \          j  j    i        f  the  work  done 

\  =  the  heat  added  -f  < 
internal  energy  J  \  on  the  system. 

This  is  in  fact  merely  a  generalization  (applied  to  a  system  of 
bodies  instead  of  to  a  body)  of  the  statement  of  §292  that  the 
heat  added  to  a  body  =  the  increase  in  internal  energy  +  exter- 
nal work  done  by  the  body.  The  essential  idea  is  that  the  energy 
added  must  all  be  accounted  for — no  part  of  it  is  lost. 

343.  Isothermal  Processes. — Any  process  or  change  of  condi- 
tion in  a  system  which  takes  place  without  change  in  temperature 
is   an  isothermal  process.     We  must  distinguish  between   an 


282  HEAT 

isothermal  process  and  an  isothermal  curve  for  a  substance. 
Suppose  the  substance  is  in  the  gaseous  state,  then  we  have  seen 
(§281)  that  at  a  given  absolute  temperature  T  the  possible 

pressures  and  volumes  are  given 
very  approximately  by  PV =RT, 
the  isothermal  curves  being  rec- 
tangular hyperbolae.  A  gas 
having  the  pressure  and  volume 
determined  by  this  equation,  at 
the  given  temperature,  would, 
if  confined  in  a  cylinder  with 
movable  piston,  be  in  equilib- 
V  rium,  that  is,  the  piston  would 


Fio.    215.—  Isothermal    equilibrium       no^    move.        All    the   Conditions 
curve,  and  curves  of  reversible  isothermal         ,    .  .        .      . 

determined    by    the    equation 


PV  =  RT  or  the  corresponding 

isothermal  equation  for  a  substance  not  a  gas,  are  equilibrium 
conditions,  the  pressure  being  the  equilibrium  pressure  cor- 
responding to  the  given  volume  and  temperature.  It  is  evident 
that  to  make  a  substance  change  its  condition  (T  constant) 
the  confining  pressure  must  be  changed  from  the  equilibrum 
pressure,  increased  if  it  is  desired  to  compress  the  gas, 
diminished  if  the  gas  is  to  be  allowed  to  expand.  If  the 
change  in  pressure  is  very  slight  the  change  in  condition  is  slow; 
if  the  pressure  is  kept  continuously  slightly  different  from  the 
equilibrium  pressure  given  by  PV  =  RT=  constant,  the  gas  will 
pass  through  a  series  of  conditions,  in  this  case  isothermally 
The  volumes  of  the  gas,  and  the  corresponding  pressures  exerted 
by  the  piston  upon  it,  plotted  on  the  PV  diagram  (Fig.  215)  give 
the  dotted  curve  just  above  and  below  the  isothermal  curve  for 
the  same  temperature  and  by  making  the  process  slow  enough 
the  dotted  curve  representing  it  may  be  made  to  approach  as 
near  as  we  wish  to  the  equilibrium  curve.  We  have  seen  that 
.the  work  done  upon  the  gas  during  isothermal  compression  is 
equal  to  the  area  under  the  isothermal  curve  between  the  extreme 
ordinates,  and  from  the  law  of  conservation  of  energy  it  follows 
that,  neglecting  the  change  in  internal  energy  with  volume,  which 
we  have  seen  (§292)  is  small,  the  equivalent  of  the  work  done 
upon  the  gas  must  be  taken  away  as  heat  in  order  to  keep  the 


THERMODYNAMICS  283 

temperature  constant.  In  a  similar  way,  during  isothermal 
expansion  heat  must  be  added. 

344.  Adiabatic  Processes. — Any  process  carried  out  in  such  a 
way  that  no  heat  is  allowed  to  enter  or  leave  the  system  during 
the  change,  is  called  an  adiabatic  process.  The  same  distinction 
as  before  exists  between  a  process  and  a  curve,  an  adiabatic 
curve  determining  a  series  of  equilibrium  conditions. 

Through  every  point  on  the  PV  diagram  one  adiabatic  and  one 
isothermal  curve  will  pass,  the  adiabatic  being  everywhere 
steeper  than  the  isothermal,  because,  since  no  heat  is  added, 
the  temperature  of  the  gas  will  fall  as  it  expands  and  does  work. 
Conversely,  a  substance  has  its  temperature  raised  by  adiabatic 
compression,  the  heat  equivalent  of  the  work  done  remaining  in 
the  substance,  the  work  done  being  always  represented  by  the 
area  under  the  adiabatic  curve  between  the  extreme  ordinates. 
The  difference  between  isothermal  and  adiabatic  compression 
may  easily  be  illustrated  by  the  use  of  a  good  bicycle  pump,  a 
slow  compression  being  almost  isothermal,  the  heat  passing  off 
as  it  is  generated,  through  the  metal  walls  of  the  cylinder,  while 
quick  compression  warms  the  gas  and  cylinder  considerably, 
as  will  be  evident  to  the  touch.  Since  it  is  impossible  ever 
entirely  to  eliminate  loss  of  heat  by  conduction,  convection,  or 
radiation,  quick  changes  of  volume  will  in  general  be  more  nearly 
adiabatic  than  slow.  The  compressions  of  sound  waves  are 
adiabatic  for  this  reason. 

The  equation  of  an  adiabatic  curve  of  a  perfect  gas,  in  the  PV 
diagram,  is 

PV*  =  constant 

K  being  the  ratio  of  the  two  specific  heats,  — •    The  same  equa- 

sv 

tion,  having  the  same  meaning,  also  holds  very  approximately  for 
real  gases  which  closely  follow  Boyle's  law;  and  even  for  C02 
which  departs  very  considerably  from  Boyle's  law,  the  adiabatic 
curve  is  given  by  the  same  form  of  equation,  though  K  is  not  in 

this  case  the  ratio  — • 
sv 

346.  The  Equation  of  an  Adiabatic. — Consider  1  gram  of  a  gas,  which 
obeys  Boyle's  law  at  least  approximately,  confined  in  a  cylinder  with  a 
movable  piston. 


284 


HEAT 


Let  V—the  initial  volume  of  gas. 

P  =  corresponding  pressure  of  the  gas. 

21  —  temperature  of  the  gas. 

On  the  PV  d  agram,  Fig.  216,  this  condition  will  be  represented  by  the 
point  (a).  Let  T  be  the  isothermal  curve  (temperature  T)  through  this 
point.  Let  the  piston  be  moved  so  as  to  compress  the  gas  an  amount  ab  =  &V. 
If  no  heat  is  allowed  to  enter  or  leave  the  gas  during  this  compression,  then, 
by  definition,  the  pressure  and  volume  after  compression  determine  a  point 
(d)  on  the  adiabatic  through  (a).  If,  on  the  other  hand,  the  temperature 
is  maintained  constant  during  compression,  then  the  final  condition  is  the 
point  (c).  In  the  first  case  the  work  done  on  the  gas  is  measured  by  the 
area  (adef),  and  it  is  evident  that  the  less  A  V,  the  more  nearly  will  this 
area  be  equal  to  the  area  (dbef).  Hence  for  small  values  of  A  V,  we  can 


T+AT 


Fia.  216. — A  small  adiabatic  change  analyzed  into  the  equivalent  "volume  constant"  and 
"pressure  constant"  changes. 

substitute  for  the  direct  compression  (a,  d)  the  steps  (a,  6)  and  (b,  d),  and 
it  follows  from  the  definition  of  an  adiabatic  that 

the  heat  given  out  in  step  (a,  6)=»the  heat  added  in  step  (6,  d) 
or 

spAT=«y(  A5P+  AT)  (1) 

But  the  change  in  pressure  of  a  gas  confined  at  constant  volume  is  approxi- 
mately proportional  to  the  change  in  temperature;  hence,  from  (1) 

SP     AJF-H  AT     AP  " (2) 

s7~       AT       "  A'P=* 

Sp 

where  K  denotes  the  ratio  — 

8V 

But,  sifice  (c)  is  on  the  isothermal  through  (a),  and  AF  is  negative 

(P+  A'P)(F+  AF)=PF, 
(see  §223)  or 

0  (3) 


THERMODYNAMICS 


285 


and,  substituting  from  (2)  for  A'P, 

VAP 


or,  for  infinitesimal  changes,  we  have 


(4) 


and,  integrating, 

PF*  =  constant. 

346.  Adiabatic  Elasticity  of  a  Gas.  —  The  modulus  of  volume  elasticity  of 
a  gas  has  been  defined  in  §169  as 


From   equation  (4)  of  §345  we  see  at  once  that  for  an  adiabatic  change 
of  volume 


BA*    -- 

347.  Cyclic  Operations. — 

A  cyclic  operation  or  cycle,  is  a 
process  or  a  series  of  processes  so 
arranged  that  the  system  under- 
going these  changes  is  finally 
brought  back  to  its  initial  condi- 
tion. On  the  PV  diagram  any 
closed  curve  would  evidently 
represent  a  cycle.  Any  such 
cycle  may  be  divided  into  an 
expansion  and  a  contraction 


.  V~KP 

Ad 


Fia.  217. — Curve  representing  a  cyclic 
operation. 


(Fig.  217),  and  the  area  under  the  curve  ABC  represents  the 
work  done  by  the  substances  in  the  expansion  ABC,  while  the 
area  under  the  curve  CD  A  represents  the  work  done  on  the 
substance,  during  the  compression  CD  A.  The  net  work,  in 
this  case  done  by  the  substance,  is  evidently  the  area  enclosed 
by  the  curve  A  BCD.  If  the  cycle  were  described  in  the  opposite 
sense,  ADCB,  the  same  amount  of  net  work  would  be  done  on 
the  substance.  These  conclusions  are  entirely  general. 

348.  Reversible  Processes  and  Cycles. — Any  process  is  defined 
as  reversible  if  it  can  be  made  to  take  place  in  the  opposite  sense 
by  an  infinitesimal  change  in  the  conditions,  or,  what  is  the  same 
thing,  if  the  curve  representing  the  process  (§343)  lies  infinitesi- 
mally  near  an  equilibrium  curve.  For  example,  to  make  an 
isothermal  process  reversible,  the  pressure  during  expansion 


286  HEAT 

must  always  be  infinitesimally  near  but  less  than,  and  during 
compression,  infinitesimally  near  but  greater  than,  the  equilib- 
rium pressure  given  by  PV  =  RT,  and  the  flow  of  heat  must  take 
place  under  an  infinitesimal  temperature  gradient,  that  is  to  a 
body  whose  temperature  is  dT  lower  than  that  of  the  gas,  or  from 
a  body  whose  temperature  is  dT  greater  than  that  of  the  gas. 
Under  these  conditions  infinitesimal  changes  in  P  and  T  will 
cause  the  process  to  be  described  in  the  opposite  direction.  A 
cycle  will  be  reversible  if  it  is  entirely  made  up  of  reversible 
processes. 

349.  The   Carnot   Cycle. — Carnot's   Cycle,  Fig.  218,  is  made 

up  of  two   isothermal  and  two  adiabatic  processes  so  chosen 

P  that  the  initial  and  final  states 

are  the  same.  Given  a  material, 
called  the  "working  substance," 
conveniently  (though  not  neces- 
sarily) a  gas,  inclosed  in  a  cylinder 
with  non-conducting  walls  and 
piston  and  a  good  conducting  bot- 
tom (Fig.  219),  together  with  a 
body  (1)  of  very  large  heat  capac- 
~  771 ~ ; ity.  at  temperature  Tlt  a  non- 

Fio.  218.— Carnot  cycle.  J '  .         *  l9 

conducting  stand  (o),  and  a  sec- 
ond body  (2)  of  large  heat  capacity  at  temperature  773,  the  Car- 
not cycle  may  be  carried  out  as  follows: 

1.  Given  the  working  substance  initially  in  the  condition 
pi>  vi>  Ti>  (A>  FiS-  218)  P^ce  the  cylinder  on  (1)  and  allow  the 
gas  to  expand  slowly  to  the  condition  P2,  V2t  Tlt  absorbing  heat 
by  conduction  from  (1)  during  the  process.     If  done  slowly  the 
process  will  be  isothermal  and  reversible. 

2.  Place  the  cylinder  on  the  insulated  stand  S,  and  allow  the 
working  substance  to  expand  adiabatically  (and  reversibly)  to 
the  condition  P3,  Vs,  Tv 

3.  Place  the  cylinder  on  the  refrigerator  (2)   and  compress 
isothermally  to  the  condition  P4,  74,  T2,  heat  being  given  off 
during  the  process  to  body  2.     This  will  also  be  reversible  if  the 
compression  is  slow. 

4.  Place   the   cylinder   again   on   the   insulating   stand   and 
compress  adiabatically  to  the  initial  condition. 


THERMODYNAMICS 


287 


According  to  §347  the  net  work  (W)  done  by  the  gas  when  the 

cycle  is  described  in  this  sense  is  represented  by  the  area  ABCD. 

Let  ^t=heat  taken  in  at  temperature  Tl  in  mechanical  units. 

#2  =  heat  given  out  at  temperature  T2  in  mechanical  units. 

Then  according  to  the  first  law 

W  =  Hl-H2 

If  the  cycle  were  carried  out  in  the  reverse  order,  then — 
H\  =  heat  given  out  at  temperature  Tl 
H'i  =  heat  taken  in  at  temperature  T2 
W'  =  work  done  on  the  gas  during  one  cycle 
and  again 


Working 
Substance 


r, 

Heater  of 
Large  Capacity 
for  Heat 

s 

Non 
Conducting 
Stand 

Refrigerator  of 
Large  Capacity 
for  Heat 

Fio.  219. — Carnot  "engine";  a  device  for  using  the  Carnot  cycle. 

In  order  that  the  cycle  may  be  reversed  it  is  necessary,  as  we 
have  seen,  that  the  heat  flow  should  take  place  with  infinitesimal 
temperature  gradient,  and  the  pressures  always  be  infinitely  near 
equilibrium  pressures.  The  first  condition  can  be  satisfied  as 
near  as  we  wish  by  making  the  isothermal  transformations  slow 
enough  and  the  second  condition  by  properly  altering  the  force 
on  the  piston.  The  process  which  we  have  described,  which 
enables  us  by  means  of  a  reversible  Carnot  cycle  to  get  mechanical 
work  from  heat,  is  called  an  ideal  heat  engine.  We  are  not  con- 
cerned at  present  with  the  mechanical  construction  of  such  an 
engine;  the  essential  characteristic  is  the  reversible  Carnot  cycle, 
and  one  ideal  engine  differs  from  another  only  in  the  material 
used  as  working  substance,  and  in  the  temperatures  and  pressures 


288  HEAT 

at  which  it  works.  An  engine  working  between  the  tempera- 
tures 7\  and  T2  in  which  the  flow  of  heat  to  and  from  the  working 
substance  took  place  under  finite  temperature  gradients,  or  in 
which  the  forces  on  the  piston  were  not  properly  adjusted  during 
expansion  and  compression,  or  both,  would  be  irreversible. 

350.  Efficiency  of  an  Engine. — The  efficiency  of  an  engine  is 
the  ratio  of  the  mechanical  work  obtained  to  the  heat  taken  in 
by  the  working  substance,  during  one  cycle.  For  the  Carnot 
cycle  this  is 

W    H.-H, 
~H~    ff, 

The  efficiency  gives  the  fraction  of  the  heat  taken  in  which  is 
transformed  into  mechanical  work. 

361.  The  Second  Law  of  Thermodynamics. — The  second  general 
principle  of  thermodynamics  was  first  formulated  independently 
by  Clausius  (1850)  and  Kelvin  (1851)  in  equivalent  but  different 
forms,  as  follows: 

It  is  impossible  for  a  self-acting  machine  to  convey  heat  from 
one  body  to  another  at  a  higher  temperature  (Clausius). 

It  is  impossible  by  means  of  any  continuous  inanimate  agency 
to  derive  mechanical  work  from  any  portion  of  matter  by  cooling 
it  below  the  lowest  temperature  of  its  surroundings  (Kelvin). 

These  are  equivalent  axioms  or  assumptions,  which  it  is  impos- 
sible to  prove  directly,  but  which  are  to  be  accepted  as  a  basis  of 
reasoning  until  some  deduction  from  them  is  found  to  contradict 
fact.  No  such  contradiction  has  ever  been  found.  The  second 
law  recognizes  and  expresses  a  certain  natural  tendency  of  events, 
for  example  the  tendency  of  heat  to  flow  down  a  temperature 
gradient — of  a  compressed  gas  to  expand.  Stated  in  another 
way  it  expresses  the  easily  accepted  generalization  that  natural 
processes — that  is,  processes  which  take  place  without  assistance 
or  control,  are  in  general  irreversible,  as  we  have  used  the  term. 

352.  Carnot's  Theorem. — We  can  now  prove  an  extremely 
important  theorem,  which  was  first  stated  in  1824  by  Carnot  as 
follows: 

The  efficiency  of  all  reversible  engines  taking  in  and  giving  out 
heat  at  the  same  two  temperatures,  is  the  same,  and  no  irreversible 
engine  working  between  the  same  two  temperatures  can  have  a 
greater  efficiency  than  this. 


THERMODYNAMICS  289 

Garnet's  proof  of  this  theorem  was  incorrect,  being  based  on 
the  caloric  theory  of  heat.  As  given  by  Clausius  and  Kelvin  it 
is  a  necessary  consequence  of  the  second  law.  First  consider  any 
two  reversible  ideal  engines,  E  and  E'  working  between  the 
temperatures  Tl  and  T2,  and  let  E'  run  backward.  Let  Hl  and 
H2  be,  as  before,  the  heat  taken  in  and  given  out  by  the  forward- 
running  engine,  and  H\  and  H'2  the  heat  given  out  and  taken 
in  by  the  engine  running  backward.  Also  let  the  engines  be  so 
connected  mechanically,  and  of  such  a  size  or  speed  that  the 
work  done  by  the  forward-running  engine  just  suffices  to  operate 
the  backward-running  engine.  Finally,  let  us  assume  for  the 
moment  that  the  efficiency  of  the  forward-running  engine  is 
greater  than  the  efficiency  of  the  backward-running  one.  Then 


from  the  inequality  of  efficiencies,  and 

W^Hi-H^H'i-H'^W  (2) 

from  the  equality  of  the  work  done  by  and  on  the  engines, 
respectively.    Hence  from  (1)  and  (2) 


or 

and  from  (2)  H2<H', 

Hence,  the  net  result  is  that  an  amount  of  heat  equal  to 


is  transferred  from  the  body  at  the  lower  temperature  !T3  to  the 
body  at  the  higher  temperature  Tlt  without  the  necessity  of 
doing  any  work.  This  violates  the  Clausius  statement  of  the 
second  law  —  hence  we  conclude  that  e  cannot  be  greater  than  ef. 
If  we  run  engine  E'  forward  and  E  backward,  we  can  prove  by 
exactly  similar  reasoning  that  e'  cannot  be  greater  than  e,  hence 
it  follows  that  e  =  e',  which  proves  the  first  part  of  the  theorem. 
If  engine  E  is  an  irreversible  engine,  then  we  can  prove  exactly 
as  above  that  e,>  cannot  be  greater  than  e',  but  since  E  cannot 
be  reversed,  we  cannot  prove  that  e'  cannot  be  greater  than  e»>. 
10 


290  HEAT 

Hence  all  we  can  say  is  that  e<r  is  equal  to  or  less  e, 


which  proves  the  second  part  of  the  theorem. 

353.  Thermodynamic  Scale  of  Temperature. — Since  the  ef- 
ficiency of  a  reversible  engine  is  independent  of  the  working 
substance  and  the  pressures  used,  it  follows  that  the  efficiency 
can  depend  only  on  the  two  temperatures  between  which  the 

H  —H 
engine  works.    If  — jf — '  depends  only  on  the  temperatures  Tl 

TJ 

and  T2,  then  —  also  depends  only  on  the  temperatures.     This 
"i 

fact  led  Lord  Kelvin  to  suggest  a  new 

scale  of  temperature,  which,  since  it 
depends  on  Carnot's  theorem  and  is  in- 
dependent of  the  properties  of  any  par- 
ticular substance,  is  called  the  absolute 
thermodynamic  scale  of  temperature.  Ac- 
cording to  this  scale,  any  two  tempera- 
tures are  to  each  other  as  the  heat  taken  in 
and  given  out  by  a  reversible  engine  de- 
scribing a  Carnot  cycle  between  these  two 
temperatures.  That  is,  if  we  call  6l  and 
02  the  thermodynamic  measure  of  two 

We  still  have  to  determine  the  size  of 


Fio.  220. — Thermodynamic 
temperature  scale;  difference 
in  temperature  proportional 
to  work  done  (area). 


02        #2 

temperatures,  ~/~jj' 

the  degree,  which  is  done,  as  in  the  case  of  the  hydrogen  scale 
by  assuming  100°  between  the  freezing-  and  the  boiling-point  of 
water,  Fig.  220.  This  amounts  to  dividing  the  area  (Fig.  220) 
between  the  0°  and  100°  isothermals  and  any  two  adiabatic 
curves  as  shown,  into  one  hundred  equal  parts. 

We  have  then 
and  by  definition 

hence 


and 


io 


100 


THERMODYNAMICS  29  1 

that  is,  the  thermodynamic  temperature  of  0°  centigrade  is 
obtained  by  dividing  the  heat  given  out  by  the  reversible  engine 
at  0°C.  by  T^  of  the  work  done  in  the  cycle  from  100°C.  to  0°C. 
Similarly  for  any  other  temperature  0  we  have 

0  =  0    *--     -*- 

0  #0    #100-#0  (from  above) 

5100 

which  can  be  interpreted  in  the  same  manner.  Absolute  thermo- 
dynamic zero  is  the  temperature  at  which  no  heat  is  given  out  by 
a  reversible  engine  working  with  this-  as  its  lower  limit.  From 
the  definition  we  see  at  once  that  differences  of  thermodynamic 
temperatures  are  proportional  to  the  work  done  by  an  ideal 
engine  working  between  these  limits,  and  hence  to  area  on  the 
PV  diagram  (Fig.  220).  We  may  also  obtain  an  expression  for 
efficiency  in  terms  of  thermodynamic  temperature,  namely 


and  since  experiment  shows  that  6  and  T  are  practically  the 
same,  we  have,  approximately 


If  there  were  a  reversible  ideal  engine  actually  available  the 
method  of  determining  thermodynamic  temperatures  would  be 
first  to  work  the  engine  between  boiling  water  and  melting  ice 
and  determine  the  amount  of  work  it  could  do,  then  to  work  it 
between  zero  and  a  source  of  available  temperature,  such  as  a 
large  tank  of  water,  and  adjust  the  temperature  of  the  tank  until 
the  work  done  was  T^  of  the  amount  done  from  100°  to  0°C. 
The  tank  would  then  be  at  +1°G.  on  the  thermodynamic  scale. 
A  similar  process  would  determine  other  temperatures. 

354.  Comparison  of  Thermodynamic  and  Hydrogen  Scale.  —  The  thermo- 
dynamic scale  is  entirely  distinct  from  the  hydrogen  scale,  and  if  it  is  to  be 
adopted  as  the  standard  we  must  have  either  a  practicable  way  of  measur- 
ing in  terms  of  it,  or  a  way  of  comparing  other  scales  with  it. 

It  can  be  proved  theoretically  that  the  temperature  indicated  by  a  gas 
thermometer  operating  with  a  perfect  gas  would  agree  exactly  with  the 
thermodynamic  temperature  as  defined  above,  using  the  perfect  gas  as  the 


292  HEAT 

working  substance.  Unfortunately  there  is  no  perfect  gas  available  for 
use  in  a  thermometer;  but  as  we  have  already  pointed  out,  the  properties 
of  real  gases  approach  those  of  a  perfect  gas  as  their  densities  approach 
zero.  Accordingly,  if  a  given  gas,  for  example  hydrogen,  is  used  in  a 
thermometer  at  several  densities,  and  the  corresponding  temperature  scales 
are  compared,  the  scale  obtained  by  extrapolating  from  these  to  a  condition 
of  zero  density,  will  agree  with  the  thermodynamic  scale.  Moreover,  real 
gases  differ  from  perfect  gases  in  several  important  ways,  namely: 

perfect  \  gases  obey  the  law  PV  -RT  (       exaotlyt  , 
real       J  |^  approximately 

the  internal  work  of  free  expansion  is  <  >  for  <  ^      .     >  gases 

\  not  zero  J         \    real     /  6 

and  the  specific  (     constant     }  .      (  perfect  } 

heat  at  constant  pressure  s  \  not  constant  J  \  real  J  * 
Hence,  by  measuring  the  pressure  and  volume  of  a  real  gas  at  various  con- 
stant temperatures,  by  performing  the  porous  plug  experiment  (§296)  with 
it,  and  by  measuring  its  specific  heat  under  various  conditions,  it  is  possible 
to  determine  its  "degree  of  imperfection,"  so  to  speak,  and  thence  the 
relation  between  its  constant  volume  temperature  scale  and  the  thermo- 
dynamic scale.  There  is  still  lacking  much  information  concerning  the 
properties  of  real  gases,  especially  concerning  the  internal  work  of  free 
expansion,  due  to  molecular  forces.  Nevertheless  the  reduction  to  the 
thermodynamic  scale  is  known  with  considerable  accuracy  for  both  hydrogen 
and  nitrogen,  as  given  in  Table  17. 

TABLE  17 

CORRECTIONS  FOR  CONSTANT  VOLUME  THERMOMETER  SCALES 

M.  HG. 


Nitr0gen'  Hydrogen. 

-  240  +0.18 

-  200  +0.62  +0.06 

-  150  +0.26  +0.033 

-  100  +0.10  +0.010 

-  50  +0.03  +0.005 
+     10                                -0.002                               -0.000 
+     40                                -0.006                               -0.001 
+     70                                -0.004                               -0.001 
+  200                                +0.04                                +0.004 
+  450                                +0.19                                +0.02 
+  1000                                 +0.70                                +0.07 
+  1200                                +1.00 

It  is  evident  that  for  moderate  temperatures  and  approximate  work  the 
thermodynamic  and  hydrogen  scales  may  be  considered  identical. 

355.  Entropy.  —  Returning  now  to  the  Carnot  cycle  we  see  that  as  a  result 
of  the  definition  of  temperature,  we  have 

Hj     6±          H^     HI 
Hi  "  61  0,  "  Ol 


THERMODYNAMICS 


293 


That  is  to  say,  the  ratio  of  the  heat  taken  in  (or  given  out)  to  the  tempera- 
ture at  which  it  is  taken  in  (or  given  out)  is  the  same  for  all  isothermal 
changes  between  any  two  adiabatica.  This  fact  suggested  to  Clausius  that 

TT 

the  quantity  -«-  is  the  change  in  a  certain  property  of  the  working  sub- 
stance, a  property  which  remains  constant  during  any  (reversible)  adiabatic 
process  but  changes  when  the  substance  passes  from  one  adiabatic  to 
another.     This  property  Clausius  named  "en- 
tropy," and  it  is  exceedingly  important. 

In  order  to  obtain  a  definite  numerical  meas- 
ure for  the  entropy  of  a  body  in  every  physical 
condition,  we  must  select  some  condition,  repre- 
sented by  a  point  on  the  PV  diagram,  as  an 
arbitrary  zero  of  entropy,  just  as  we  select  the 
sea  level  as  the  zero  from  which  to  measure 
heights  and  depths.  Suppose  P  (Fig.  221)  is 
the  adopted  zero,  then  the  entropy  of  any  other 
state  P'  is  obtained  by  measuring  the  heat  taken 
in  (or  given  out)  in  passing  from  P  to  P'  by  a 
reversible  path.  The  simplest  path  is  by  the 
adiabatic  PN  and  the  isothermal  0.  If  H  is 
the  heat  taken  in  in  passing  from  N  to  P't  then  the  entropy  of  P'  with  respect 

JPf  H 

to  P,  which  we  shall  represent  by  Srp1  would  be  equal  to  -j-'    If  P'  were 

reached  by  another  reversible  path  involving  portions  of  several  adiabatics 
and  isothermals,  and  quantities  of  heat  Hlt  H2,  H9  •  •  •  were  taken  in  (or 
given  out)  at  the  temperatures  $lt  02,  0,,  •  •  •  then 
P' 

SD  =— -  +  — -+— -=-    >  —  •    If  any  of  the  quantities  of  heat  Hlt  #.  •  •  • 
f        6        0        0        ^— J  0 


FIG.  221 . — Arbitrary  zero  of 
entropy,  P;  Entropy  of  P'  de- 
termined by  adiabatic-isother- 
mal  change  from  P. 


were  given  out  by  the  body  they  should  be  taken  with  the  minus  sign  in 
the  summation.  It  is  evident  that,  defined  in 
this  way,  every  state  has  a  definite  entropy. 

356.  Entropy  and  Reversible  Cycles  in  Gen- 
eral.— We  have  seen  that  in  passing  around  a 
Carnot  cycle  the  entropy  of  the  working  sub- 
stance was  not  changed.  This  result  may  be 
extended  to  include  any  reversible  cycle,  repre- 
sented by  the  closed  curve  in  Fig.  222.  By 
y  drawing  a  series  of  adiabatics  across  this  and 
connecting  these  around  the  edge  by  a  series 

^.TS-SS  caTt  °f  -'"  ^  «  f™>  *•  - *•*  *• 

cycles.  given  cycle  may  be  broken  up  into  a  series  of 

Carnot  cycles  the  sum  of  whose  areas  will  ap- 
proach the  area  of  the  given  cycle  as  a  limit,  as  their  number  is  increased. 
Furthermore,  the  heat  taken  in  along  the  isothermal  steps  A  A',  BB't  CC', 


294  HEAT 

DD'y  etc.,  is  equal  in  the  limit  to  the  heat  taken  in  moving  along  the  curve 
A— D,  for  the  difference  would  be  represented  by  the  sum  of  the  triangular 
areas,  which  is  zero  in  the  limit.  Hence,  since  for  each  elementary  Carnot 
cycle  we  have — 


we  have  for  the  whole  cycle    ^  — -  —  ^  —  «  0 

or,  if  we  give  the  negative  sign  to  heat  H9  leaving  the  system,  this  becomes, 


f- 

when  the  number  of  elementary  cycles  has  become  infinite.    This  shows  us 
that  all  reversible  paths  between  two  conditions  involve  the  same  change 

dH 

in  entropy,  or  that  —  is  a  perfect  differential. 

357.  Increase  in  Entropy. — If  an  amount  of  heat  H  flows  from  one  body 
at  a  temperature  0,  to  another  at  a  lower  temperature  0,,  the  entropy  of 

TT 

the  hot  body  is  decreased  by  an  amount  —   and  that  of  the  cooler  body  ia 

H  /  1        1 

increased  by  —  •  Evidently  in  all  cases  of  conduction,  dS**H(  — — 

0*  \*t      0i 

is  positive,  or  the  entropy  of  the  two  bodies  is  increased. 

It  can  also  be  proved  that  other  "natural"  processes  such  as  free  or 
unbalanced  expansion,  diffusion,  the  falling  of  bodies  in  obedience  to 
gravitation,  and  the  production  of  heat  from  mechanical  energy  by  friction, 
all  involve  an  increase  in  entropy.  These  processes  are  also  all  irreversible, 
and  they  all  tend  to  a  more  uniform  condition  as  regards  temperature, 
pressure  and  the  velocities  of  bodies  and  of  molecules.  Hence  it  is  a 
reasonable  extension  of  our  ideas  to  say  that  all  natural  processes  are  irre- 
versible and  lead  to  an  increase  in  entropy,  and  to  associate  the  increase  of 
entropy  with  increase  in  the  uniformity  of  physical  conditions.  All  natural 
changes  seem  to  be  tending  to  a  condition  of  maximum  uniformity.  This 
additional  hypothesis,  that  natural  processes  always  lead  to  an  increase  in 
entropy,  is  the  basis  for  the  discussion  of  problems  of  chemical  and  physical 
equilibrium,  such  as  the  equilibrium  of  a  liquid  with  its  vapor,  of  a  solid 
with  its  liquid,  or  of  different  chemical  compounds  with  each  other. 

We  have  seen  that  for  any  irreversible  cycle  during  which  heat  is  taken 
in  and  given  out  at  temperatures  6l  and  02, 


from  which 


THERMODYNAMICS 


295 


or 


or 


r 
J 


dH 


0 


So  that  in  this  general  case  all  we  can  prove  from  the  second  law  is  that 
there  may  be  an  increase  in  entropy;  and  by  an  extension  of  this  reasoning 
it  may  be  proved  that  no  irreversible  process  can  lead  to  a  decrease  in  entropy. 

358.  Reciprocating  Steam  Engines. — The  ordinary  reciprocat- 
ing steam  engine,  one  type  of  which  is  shown  in  Fig.  223,  ia  the 
most  common  machine  used  to  convert  heat  energy  into  mechan- 
ical work.  In  these  engines  water  is  the  working  substance 


Water  Inlet 


Condensed  Steam 


Fio.  223. — Reciprocating  steam  engine  and  condenser  in  which  steam  is  condensed  on 

water-cooled  pipes. 


(Compare  Art.  349) ,  the  boiler  is  the  source  of  heat  at  the  higher 
temperature,  and  the  cooling  water  of  the  condenser  is  the  cooler 
body  into  which  heat  is  discharged.  Instead  of  moving  the 
cylinder  from  one  to  the  other  as  was  before  suggested,  it  is 
obviously  easier  to  conduct  the  working  substance  from  point  to 
point.  On  account  of  mechanical  difficulties  no  attempt  is 
made  to  realize  completely  the  Carnot  cycle  (Art.  349),  but  the 
actual  cycle  through  which  the  working  substance  passes  is 
of  the  form  shown  in  Fig.  224.  The  operations  are  as  follows: 


296 


HEAT 


(1)  Water   is   vaporized   in   the   boiler  at  the  temperature 
TI,  absorbing  an  amount  of  heat  Ll  per  unit  mass  (heat  of 
vaporization). 

(2)  Steam  passes  at   constant  pressure  P^  from  the  boiler 
through  the  valve  a  (Fig.  223),  into  the  cylinder  as  the  piston 
begins  its  motion  to  the  right.    Thus  the  isothermal  expansion 
at  pressure  Pt  due  to  vaporization  is  represented  by  the  line 
(A,B,),  and  this  expansion  does  an  amount  of  work  represented 
by  A,B,0,F. 

(3)  The  valve  a  closes,  and  the  saturated  steam  expands 
from  B  to  D.     This  expansion  should  be  as  nearly  as  possible 

adiabatic.  The  work  done  is 
represented  by  the  area  BDHG. 
At  C  the  valve  a  opens  to  the 
exhaust  e,  the  steam  begins  to 
escape  to  the  condenser,  and  the 
pressure  falls  quickly  from  C  to  D. 
(4)  The  piston  reverses  its  mo- 
tion at  D,  and  the  motion  to  the 
left  is  opposed  by  the  constant 
pressure  P2,  since  during  this  time 
there  is  isothermal  condensation 

of  the  steam  in  the  condenser,  at  temperature  T2.  The  temper- 
ature T2  is  fixed  by  the  cooling  water  which  is  available  for  the 
condenser.  With  a  non-condensing  engine  T2  is  necessarily 
about  373°  absolute  (100°G.).  An  amount  of  heat  L/per  unit 
mass  is  given  up  to  the  cooling  water  during  the  process  of 
condensation,  and  work  to  the  amount  DEFH  is  done  on 
the  steam. 

(5)  The  condensed  steam  is  heated  at  constant  volume  (E,A) 
and  admitted  to  the  boiler  at  A,  thus  completing  the  cycle. 
This  requires  an  additional  amount  of  heat  H. 

It  is  possible  to  arrange  a  mechanism  so  that  the  engine  as 
it  runs  will  automatically  draw  a  curve  whose  ordinates  are  pro- 
portional to  the  pressure  of  steam  in  the  cylinder,  and  whose 
abscissae  are  proportional  to  the  corresponding  volume  occupied 
by  the  steam  in  the  cylinder.  This  curve  is  very  similar  to  the 
one  of  Fig.  224,  and  is  called  an  indicator  diagram. 


F          G  H 

Fio.  224. — Ideal  (Rankine)  cycle  for  a 
reciprocating  steam  engine. 


THERMODYNAMICS  297 

EFFICIENCY  OF  ENGINES 

359.  The  work,  W,  done  per  pound  of  steam  is  evidently  repre- 
sented by  the  area  ABDE,  while  the  total  heat  taken  in  is 

W 

Z/.+H.  The  ratio  77 — .  „..  r>  where  J  is  the  mechanical  equiva- 
(Li  +  H}J 

lent  of  heat ,  is  called  the  thermal  efficiency  of  the  engine.  The  ther- 
mal efficiency  measures  the  perfection  of  the  thermal  processes 
which  the  engine  uses.  There  is,  of  course,  energy  lost  (converted 
into  heat)  by  friction  among  the  moving  parts,  so  that  the  actual 
work,  W,  which  the  engine  could  do  in  running  some  machine, 

is  always  less  than  W .    The  ratio  -^  is  called  the  mechanical 

efficiency  of  the  engine,  and  its  value  is  a  measure  of  the  mechan- 
ical perfection  of  the  engine.  The  product  of  the  two  efficiencies, 

W' 
namely,  the  ratio  7J^fj\j'  evidently  measures  the  efficiency 

of  the  engine  in  the  conversion  of  heat  into  usable  mechanical 
work,  and  this  will  be  less  than  its  thermal  efficiency. 

It  is  interesting  to  compare  the  thermal  efficiency  -= — — zr-r 

(•L/i~r  H)J 

with  the  efficiency  of  an  ideal  Carnot  engine  working  between  the 
same  temperatures  T1  and  T2. 

Since,  according  to  §354  the  constant  volume  hydrogen  scale 
and  absolute  thermodynamic  scale  of  temperature  are  practically 
identical,  the  expression  for  the  efficiency  of  an  ideal  engine 

T  —T 
becomes  e  =  — 1= — ->  and  this  is  the  maximum  efficiency  which 

any  real  engine  could  possibly  be  expected  to  approach  if  it 
works  with  a  boiler  temperature  Tl  and  a  condenser  temperature 
T2.  For  example,  with  a  boiler  at  177°G.  and  a  condenser  at 

77°C.,  e  =  -7^=22  per  cent.,  that  is  to  say,  the  ideal  engine  could 


convert  less  than  one-quarter  of  the  heat  used  into  mechanical 
work.  Table  18  gives  the  actual  thermal  efficiency  and  the 
corresponding  ideal  efficiency  for  the  best  engines  of  several 
types. 

Besides  engine  efficiencies,  the  efficiency  of  boilers,  namely, 

heat  given  to  water 

the  ratio  —   — - — : — — — -  (in  a  given  time)  is  of  course 

heat  obtained  from  fuel 


298 


HEAT 


of  equal  importance  in  the  problem  of  obtaining  mechanical  work 
from  fuel.  The  average  efficiency  of  boilers  is  60  per  cent.,  the 
maximum  80  per  cent.,  so  that,  combining  the  best  boiler  with 
the  best  engine,  the  maximum  efficiency  actually  attained  is 
about  21  per  cent. 

From  §298  the  heat  of  combustion  of  soft  coal  is  2.9  XlO11 
ergs  per  gram,  or  12,500  B.  T.  U.  per  pound,  while  (§57)  one 
horse-power  for  one  hour  equals  2.68 XlO18  ergs,  or  1.98X108 
foot  Ibs.  Since  1  B.  T.  U.  equals  778  foot  Ibs.,  it  follows  that 
the  combustion  of  1  Ib.  of  coal  liberates  energy  sufficient  to 
provide  4.8  H.  P.  for  one  hour,  whereas  the  best  boiler-engine 
combination  so  far  built  obtains  0.82  H.  P.  hour  per  Ib.  of  coal. 


TABLE  18 
EFFICIENCY  OF  STEAM  ENGINES 


Temperature 

Efficiency 

Efficiency 

of  Carnot 

• 

'* 

*' 

cycle 

Per  cent. 

Per  cent. 

Willan's  engine  (non-condensing) 

164° 

101.5° 

10.4 

14.5 

Levitt    pumping    engine    (com- 

181.6° 

37.7° 

19 

31.7 

pound). 

Levitt    pumping    engine    (triple 

191.9° 

46.7° 

20.8 

31.8 

expansion). 

Nordberg  engine  (quadruple  ex- 

206.35° 

43.1° 

25.5 

34.0 

pansion). 

360.  The  Defects  of  Real  Engines. — In  the  discussion  of  §359  we  have 
neglected  several  points  of  importance.  For  example,  the  expansion  EC 
can  never  be  strictly  adiabatic  because  the  cylinder  and  piston  must  be  of 
conducting  material.  This  leads  to  the  condensation  of  steam  in  the 
cylinder.  If  it  is  attempted  to  raise  the  temperature  T,  so  as  to  increase 
the  efficiency,  the  cylinder  condensation  is  increased.  By  using  several 
cylinders  (compound,  triple  and  quadruple),  allowing  part  of  the  expansion 
to  occur  in  each,  the  temperature  changes  in  each  cylinder,  and  hence  the 
condensation  losses  are  reduced  and  it  is  possible  to  use  higher  initial  tem- 
peratures. Further  reduction  of  condensation  loss,  and  increase  of  the 
initial  temperature  without  increase  in  the  initial  pressure,  is  accomplished  by 
superheating  the  steam,  by  passing  it,  at  constant  pressure,  through  coils  of 


THERMODYNAMICS 


299 


pipe  in  the  hot  flue  gases,  as  shown  in  Fig.  225.  It  is  then  no  longer  satu- 
rated when  it  enters  the  cylinder,  and  the  cycle  would  be  represented  by 
different  lines  on  the  PV  diagram. 

Saturated  Steam    Superheated  Steam 

Jiturated    ff 
team  to 
oerheater 


Fio.  225.— BoUer  and  superheater. 

361.  Steam  Turbines. — The  turbine  is  another  type  of  machine 
of  more  recent  development,  for  obtaining  mechanical  work  from 
the  heat  energy  of  steam,  the  essential  features  being  a  rotating 
shaft  with  properly  arranged  blades  and  fixed  nozzels  or  blades 
for  directing  the  flow  of  the  steam,  which  is  initially  at  a  high 
pressure.  Turbines  may  be  divided  into  two  general  classes. 
In  the  first  class,  called  the  " velocity"  type,  steam  is  allowed  to 
expand  at  once  to  the  final  pressure,  in  a  properly  shaped  nozzle, 
so  that  the  jets  acquire  a  high  velocity.  These  jets  impinge  on 
the  movable  blades  and  cause  them  to  rotate,  much  as  the  jets  of 
water  impinge  on  the  blades  in  certain  types  of  water  wheels 
(Fig.  104,  §204).  By  using  several  sets  of  movable  blades,  with 
fixed  passages  between  for  reversing  the  direction  of  the  steam 
jet,  the  drop  in  velocity  is  rendered  more  gradual,  and  the  speed 
of  the  turbine  shaft  need  not  be  so  great.  In  the  second  class  of 
turbines,  called  the  "pressure"  type,  shown  in  Fig.  226,  the 
steam  expands  gradually  through  a  great  many  sets  of  movable 
and  fixed  blades,  exerting  a  pressure  on  each  set  which  causes  the 
movable  blades  to  rotate.  Steam  turbines  have  certain  mechan- 
ical advantages  over  reciprocating  engines,  namely,  uniform  and 


300 


HEAT 


high  angular  velocity,  freedom  from  vibration  (hence  their 
desirability  for  use  in  steamships),  and  economy  of  space. 
Turbines  are  slightly  more  efficient  than  reciprocating  engines 
for  low  working  pressures,  but  slightly  less  efficient  at  high 


Entering  Steam 


Exhaust  To  Condenser 


Path  of  Steam  Between  Stages 


25*-- 


f«  a  *ir  *-«-^ 

If  I 


5    « 

I- 11 

o        s: 

o     .2 

"I 

FIQ.  22Q. — Steam  turbine,  pressure  type;  general  arrangement  and  detail  showing  flow  of 

steam  past  blades. 

pressures.  Hence  it  has  been  found  advantageous  to  combine 
the  two,  delivering  high  pressure  steam  to  a  reciprocating  engine 
and  allowing  the  partially  expanded  steam  to  pass  from  it  to  a 
low  pressure  turbine. 

362.  Internal   Combustion   Engines. — In   these   engines   the 
function  of  the  boiler  and  expanding  cylinder  are  combined,  the 


THERMODYNAMICS 


301 


combustion  taking  place  in  the  cylinder  itself.  The  way  in  which 
this  is  carried  out  in  the  "four  cycle"  type  of  engine  can  best  be 
understood  by  describing  the  various  stages  shown  in  Fig.  227. 
In  (1)  the  inlet  valve  is  open  during  the  entire  stroke  to  the  right, 
admitting  a  cylinder  full  of  a  proper  explosive  mixture  of  a  com 
bustible  (coal  gas,  gasoline  or  alcohol  vapor),  and  air.  In  (2) 


/      Suction  Strobe 


//       Compression  Strode 


JTJ      Expansion  Stroke 


IV       Exhaust  StroHe 
Fia.  227. — Four-cycle  internal  combustion  engine,  showing  the  four  stages  of  one  cycle 


the  valve  is  closed  and  the  return  stroke  is  taking  place;  this 
compresses  the  mixture  into  the  clearance  space  at  the  end  of 
the  cylinder,  which  is  called  the  explosion  chamber.  At  the  end 
of  the  compression  stroke  the  mixture  is  exploded,  usually  by  an 
electric  spark.  The  high  pressure  resulting  from  the  combustion 
acts  upon  the  piston  during  the  stroke  (3)  to  the  right,  while  at 
the  end  of  this  stroke  the  exhaust  valve  e  opens  and  during  (4) 


302  HEAT 

the  products  of  combustion  are  expelled  preparatory  to  beginning 
over  again  as  in  (1).  Engines  using  the  series  of  operations 
just  described  are  called  "four-cycle"  engines,  because  four 
strokes  are  necessary  to  complete  the  series.  There  are  other 
types  of  engines — notably  the  "two-cycle,"  requiring  only  two 
strokes  to  complete  the  series  of  operations,  and  the  Diesel,  in 
which  air  alone  is  compressed  and  the  fuel  is  injected  into  it, 
the  result  being  quiet  combustion,  instead  of  an  explosion  as 
in  the  four-cycle  type.  The  thermal  efficiency  of  the  best  four- 
cycle engines  is  about  30  per  cent.,  of  the  Diesel  type  about  40 
per  cent.  Aside  from  high  efficiency,  internal  combustion  engines 
have  the  further  advantages  of  compactness,  ease  of  handling, 
and  quickness  of  starting.  The  formation  of  the  proper  mixture 
of  fuel  and  air  is  the  most  troublesome  operation  in  the  running 
of  an  internal  combustion  engine,  this  being  usually  accom- 
plished in  separate  attachments  called  carburetors.  The  largest 
engines  of  the  internal  combustion  type  so  far  built  are  of 
4000  H.  P. 


References 

POINCABE,  H.     The  New  Physics. 

A  very  readable  and  authoritative  book  discussing  recent  developments. 
Chapters  on  the  various  states  of  matter,  and  on  thermodynamics  and 
the  conservation  of  energy. 

TYNDALL.     Heat  as  a  Mode  of  Motion. 

Popular  lectures  illustrated  by  experiments.  A  beautifully  clear  and 
simple  presentation. 

ENCYCLOPEDIA  BRITANICA,  llth  Edition. 

Various  articles  bearing  on  the  material  of  the  previous  section,  espe- 
cially articles  ''Heat"  and  " Calorimetry,"  by  Callendar. 

GRIFFITHS.     Thermal  Measurement  of  Energy. 

A  clear,  popular  account  of  the  development,  from  Rumford's  time  to 
1900,  of  the  idea  that  heat  is  a  form  of  energy,  and  of  experiments  based 
on  this  idea. 

EDSER.     Heat. 

A  text-book  covering  about  the  same  ground  as  the  previous  section, 
but  in  greater  detail. 

POYNTING   AND    THOMSON.      Heat. 

A  text  somewhat  more  advanced  than  Edser,  especially  as  regards  the 
mathematical^treatment. 


REFERENCES  303 

PRESTON.     The  Theory  of  Heat. 

A  valuable  reference  book,  complete  and  readable.  A  judicious  com- 
bination of  theoretical  and  experimental  treatment. 

DA  via.    Elementary  Meteorology.    An  excellent  text. 

DARLING.    Heat  for  Engineers. 

A  treatment,  largely  from  the  experimental  side,  of  various  portions  of 
the  subject  of  especial  interest  to  engineers,  for  example  temperature 
measurement,  expansion,  combustion,  conduction,  refrigeration. 

GOODENOUQH.     Principles  of  Thermodynamics. 

The  principles  of  thermodynamics  are  developed  in  this  book  with 
especial  reference  to  their  application  to  steam,  gas,  and  other  heat 
engines  and  refrigerating  machines. 

TRAVEBS.    Study  of  Gases. 

A  good  chapter  on  liquefying  processes. 

BURGESS-LE  CHATELIER.     High  Temperature  Measurements,  3rd.  Edition. 
The  best  book  in  this  field. 

DAY  AND  SOSSMAN.     High  Temperature  Gas  Thermometry.    Published  by 
Carnegie  Institution. 
An  account  of  the  most  recent  methods  and  results. 

"  Scientific  Memoirs,"  edited  by  Ames,  is  a  collection  of  reprints,  or  transla- 
tions, of  various  important  scientific  papers,  each  paper  being  preceded 
by  a  short  biographical  sketch  of  the  author.  The  following  are  of 
interest: 

AMES.     The  Free  Expansion  of  Gases. 

Contains  papers  of  Gay-Lussac  (1807),  Joule  (1845),  and  Kelvin  and 
Joule  (1853-1862),  all  bearing  upon  the  changes  in  temperature  pro- 
duced by  changes  in  density  of  gases.  The  papers  of  Kelvin  and  Joule 
contain  the  account  of  the  famous  "porous  plug"  experiment. 

MAQIE.     The  Second  Law  of  Thermodynamics. 

Contains  the  papers  of  Carnot  (1824),  Clausius  (1850),  and  Ke  vin 
(1851),  in  which  the  second  law  of  thermodynamics  was  developed. 

The  following  papers  are  also  of  special  importance: 
BRADLEY.     Physical  Review,  9,  1904,  pp.  330-387. 

Interesting  results  of  efficiency  tests  of  a  liquid  air  plant. 
WAIDNER  AND  DICKENSON.     Bull.  Bur.  Standards  3,  1907,  p.  663,  contains 

descriptions  of  thermometers  and   methods  of  use  between  0°  and 

100°C. 
BARNES.     Trans.   Intern.   Electr.   Congress,  St.  Louis,  I,  9105,  p    53.    A 

good  summary  of  the  determination  of  the  mechanical  equivalent  of 

heal  by  electrical  methods. 
WAIDNER  AND  BURGESS.     Bull.  Bur.  Standards,  I,  p.  189. 

A  full  description  of  the  methods  and  instruments  of  pyrometry. 
BOUSEPIELD  AND  BousEFiELD.     Phil.  Trans.  Royal  Soc.  of  London,  211, 

1911,  p.  199. 

On  the  specific  heat  of  water  and  the  mechanical  equivalent  of  the 

calorie.     The  latest  work,  and  also  a  summary  of  previous  results. 


304  HEAT 

Problems  in  Heat 

T  1.   Find  the  value  of  the  following  temperatures  on  the 

Centigrade  scale:  The  temperature  of  the  human  body 

(98°F.);  normal  temperature  of  a  living  room;  a  cold  day  in  winter 

(20°F.  below  zero).  Ans.  36.6°C. ;  20°C. ;  -28.9°C. 

2.  Find  the  value  of  the  following  Centigrade  temperatures  on  the  Fahren- 
heit scale:  Absolute  zero;  melting-point  of  gold;  temperature  of  sun. 

Ans.   -459.40F.;1947°F.;10832°F, 

3.  At  what  temperature  do  the  Fahrenheit  and  Centigrade  thermometers 
read  the  same?    The  Fahrenheit  twice  the  Centigrade? 

Ans.  -40°;  160°C. 

„          .  4.  A  clock  which  has  a  pendluum  made  of  brass,  keeps 

correct  time  at  20°C.;  if  the  temperature  falls  to  0°C., 

how  many  seconds  will  it  gain  or  lose  per  day?        Ans.  16.3  sees.  gain. 

6.  Steel  street  car  rails,  having  their  ends  welded  together,  are  laid  in 

concrete  so  that  it  is  impossible  for  them  to  move.    Find  the  stress  in 

the  rails  at  —  10°C.,  assuming  that  they  are  laid  when  the  temperature 

is20°C.  Ans.  About  7.5 XlO8  dynes/cm1. 

6.  Compute  the  change  in  volume  of  a  block  of  iron  3  in.  X  4  in.  X  10  in. 
if  the  temperature  changes  from  44°F.  to  116°F.  Ans.  .17  cu.  in. 

7.  A  uniform  cylinder  is  filled  with  hydrogen  under  atmospheric  pressure. 
The  piston  stands  at  a  height  of  400  cms.  at  20°C.     If  the  pressure  is 
kept  constant,  find  the  height  of  the  piston  at  the  following  tempera- 
tures: 100°C.;  300°C.;  -80°C.;  -180°C. 

Ans.  510  cms. ;  783  cms. ;  263.4  cms. ;  127  cms. 

8.  If  in  the  preceding  problem  the  same  gas  is  compressed  until  the  piston 
stands  at  a  height  of  200  cms.  and  the  volume  is  then  kept  constant, 
compute  the  pressure  for  the  temperatures  given  in  problem  7. 

Ans.  193.5;  297.0;  100., -48.2  cm.  Hg. 

9.  Beginning  at  10  atmospheres  pressure  and  2  liters  volume,  10  gr.  of 
air  has  its  pressure  so  changed  that  the  p,  v  curve  is  a  45°  straight 
line.    Discuss  the  temperature  changes  which  occur.     (Fig.  187.) 

10.  Given  10  liters  of  nitrogen  at  30°C.  and  120  atmospheres  pressure, 
what  would  be  its  volume  at  100°C.  and  200  atmospheres  pressure? 

Ans.  7.4  liters. 

11.  What  would  be  the  relative  increase  in  size  of  an  air  bubble  in  passing 
from  the  bottom  of  a  lake  20  m.  deep  where  the  temperature  is  4°C.  to 
the  top  where  the  temperature  is  20°C.  ?  Ans.  F2  -  Vt  X  3. 1 . 

_  .    .  12.  100  grams  of  silver  at   100°C.  are  dropped  in  160 

grams  of  water  contained  in  an  iron    calorimeter 

weighing  40  grams.     Temperature  of  water  initially  15°C.     Compute 

the  rise  in  temperature  of  water.  Ans.  2.8°C. 

13.  50  grams  of  a  substance  at  100°C.  are  dropped  into  100  grams  of  water 

at  4°C.     If  the  water  is  contained  in  a  copper  calorimeter,  mass  60 

grams,  and  the  temperature  of  the  water  changes  to  10°C.,  compute 

the  specific  heat  of  the  substance.  Ans.  .1403  c.  g.  s.  units. 


PROBLEMS  305 

14.  A  water  heater  will  heat  50  liters  of  water  per  minute  from  15°C.  to 
80°C.;  if  the  efficiency  is  25  per  cent.,  how  many  calories  must  be 
generated  in  the  heater  to  do  this?  Ans.  13  X 10*  cals. 

16.  How  many  liters  of  gas  will  be  required  per  minute  in  the  preceding 
problem?     Density  of  gas  at  0°C.  and  760  mm.  pressure  — .0050. 

Ans.  444  liters. 

,      .     .       16.  In  drilling  a  hole  in  a  block  of  iron,  power  is  supplied 
at  the  rate  of  .8  H.  P.  for  3  minutes.    How  much 

rfHeat  heat  i8  Produced?     If  *  of  this  heat  S°es  to  warm 

the  iron  whose  mass  is  700  grams,  find  its  change  in 

temperature.  Ans.  2555  cals;  30°C. 

17.  How  much  would  the  temperature  of  water  be  raised  by  impact  after 
falling  200  ft.  under  gravity,  supposing  that  all  the  energy  due  to  its 
motion  was  converted  into  heat.  Ans.  .14°C. 

18.  Determine  the  heat  produced  in  stopping  a  fly-wheel  of  112  Ibs.  mass 
and  2  ft.  in  radius,  rotating  at  the  rate  of  one  turn  per  second,  assuming 
the  whole  mass  concentrated  in  the  rim.  Ans.  .364  B.  T.  U. 

19  If  electrical  energy  is  12  cents  per  1000  watt  hours  and  gas  $1.15  per 
1000  cu.  ft.,  what  will  be  the  relative  cost  of  gas  heating  and  electric 
heating?  See  problem  15  for  the  density  of  the  gas. 

Ans.  Cost  of  elect.  -100  times  cost  of  gas. 

_,  .       20.  How  much  would  the  air  in  a  room  6X5X3  meters  be 

„    ^  warmed  by  the  condensation  alone  of  1  kg.  of  steam 

in  the  radiator?     What  would  it  be  if  the  room  were 

air-tight?  Ans.  19.4°C.;  27.2°C. 

21.  With  what  velocity  must  a   lead   bullet  at   50°C.  strike  against  an 
obstacle  in  order  that  the  heat  produced  by  the  arrest  of  its  motion, 
if  all  produced  within  the  bullet,  might  be  just  sufficient  to  melt  it? 

Ans.  335  m/sec. 

22.  How  much  steam  at  150°C.  must  be  added  to  1  kg.  of  ice  at  -  10°C. 
to  give  nothing  but  water  at  0°C.?  Ans.  128.5  grams. 

23.  What  is  the  relative  humidity  of  air  at  30°C.  if  the  dew  point  is  found 
to  be  10°C.?  Ans.  28.7  per  cent. 

24.  How  much  heat  would  be  required  to  convert  1  gm.  of  water-substance 
from  liquid  at  0°C.  to  vapor  at  150°C.  under  1  atmosphere  pressure? 

Ans.  663  cals. 

25.  Compute  the  "external"  part  of  the  heat  of  vaporization  of  water  at 
100°C.  Ans.  40.2  cals. 

26.  Carry  a  mass  of  substance  across  the  triple-point  diagram  as  shown  on 
page  253,  explaining  just  what  happens  at  the  different  points.     Do 
this  for  both  constant  pressure  and  constant  temperature  following  the 
dotted  line  i. 

27.  If  it  is  desired  to  heat  CO2  at  constant  volume  in  a  closed  tube  and  have 
the  substance  pass  through  the  critical  point,  what  portion  of  liquid 
and  vapor  must  there  be  at  20°C.  initially? 

Ans.  About  4£  parts  vapor  to  1  of  liquid. 

20 


306  HEAT 

28.  Some  ether  is  poured  into  a  bottle  containing  air  at  atmospheric  pressure 
and  the  bottle  quickly  corked;  upon  shaking  the  bottle  and  removing 
the  cork  a  "pop"  is  heard.  Explain. 

jjeat  29.  The  walls  of  a  certain  refrigerator  have  an  area  of 

•   Co  d     ti  15,000cm.*,  and  are  made  of  cork  3  cm.  thick     Find 

out  how  much  ice  may  be  expected  to  melt  in  one  day 

if  the  outside  temperature  is  86°F.  Ans.  21  kg. 

30.  The  top  of  a  steam  chest  containing  steam  at  atmospheric  pressure 
consists  of  a  slab  of  stone  61  cm.  long,  50  cm.  broad  and  10  cm.  thick. 
The  top  being  covered  with  ice,  it  was  found  that  48  kilos  were  melted 
in  39  minutes.     What  is  the  conductivity  of  the  stone? 

Ans    .0054  c.g.s.  units. 

31.  One  end  of  a  copper  bar  4  sq.  cm.  in  cross-section  and  80  em.  long,  is 
kept  in  steam  under  one  atmosphere  pressure  and  the  other  end  in  con- 
tact with  melting  ice.     How  many  grams  of  ice  will  be  melted  in  1 0  min. ? 
Neglect  loss  due  to  radiation.  Ana.  34.3  grams. 

32.  How  much  anthracite  coal  must  be  burned  to  make  up  for  the  loss  of 
heat  due  to  conduction  for  one  day  through  a  glass  window  3  mm.  thick 
and  having  an  area  of  3  square  meters,  supposing  the  air  in  the  room  to  be 
at  temperature  25°C.,  and  the  outside  air  at  -20°C     What  important 

point  has  been  neglected?  Ans.  257  Ibs. 

Rad'at'on         ^"  ^  ^ack  radiator  2  square  meters  in  area,  is  in  a  room 

whose  walls  are  at  temperature  of  18°C. ;  if  the  radiator 

is  at  100°C.  at  what  rate  does  the  room  gain  heat?    The  constant  a  of 

Stefan's  law,  Q=>sT',  is  5.6xlO~ia  watts  per  square  centimeter. 

Ans.  326  cals./sec. 

34.  If  the  temperature  of  a  furnace  is  measured  by  allowing  the  heat 
radiated  through  a  hole  1  square  centimeter  in  area  in  the  walls  to  warm 
100  grams  of  water  placed  in  front  of  the  hole,  what  is  the  temperature 
if  the  water  rises  in  temperature  by  13°C.  in  1  minute?    Assume  that 
all  the  heat  radiated  from  the  hole  is  absorbed  by  the  water  and  also 
neglect  the  heat  radiated  back  into  the  furnace  by  the  water. 

Ans.  1660°C. 

35.  A  blackened  copper  ball  5  cm. in  diameter  is  heated  to  500  °C.  and  hung 
by  a  non-conducting  thread  in  an  exhausted  vessel.    If  the  absorbing 
power  of  the  ball  and  of  the  walls  of  the  vessel  is  .98  and  the  walls 
are  at  0°C.,  what  will  be  the  initial  rate  of  cooling  of  the  ball? 

Ans.  4.9°C.  per  sec. 

Thermo-  36.  How  many  degrees  will  dry  air  at  15°C.  rise  in  tern- 

dynamics,  perature  if  compressed    adiabatically    to    £   of  its 

volume?  Ans.  260°C. 

37.  How  much  work  would  be  done  by  air  in  expanding  adiabatically  from 
the  point  P— 760mm.  Hg.,  F-800  c.c.  to  the  point  P— 400  mm.  Hg.? 
Solve  graphically.  Ans.  337  X 10*  ergs. 

38.  Plot  a  curve  showing  the  corrections  for  a  hydrogen  thermometer. 
From  your  curve  find  the  thermodynamic  temperature  for  the  boiling 
point  of  oxygen  and  also  the  melting  point  of  lead. 

Ans.  -182.95°0.;327.013°0. 


PROBLEMS  307 

39.  What  is*  the  total  pressure  on  the  end  of  a  boiler  3  ft.  in  diameter  if  the 
temperature  of  the  water  inside  is  180°C.?  Ana.  151,000  Ibs. 

40.  A  certain  locomotive  burns  100  Ibs.  of  soft  coal  per  hour.     How  much 
work  would  the  engine  do  if  all  this  heat  were  converted  into  mechanical 
work?     In  reality  the  engine  furnishes  20  H.  P.     What  is  the  efficiency 
of  boiler  and  engine  combined?        Ans.  1097  X 10"  ft  Ibs. ;  3.6  per  cent. 

41.  If  an  engine  working  at  the  rate  of  622.4  H.  P.  keeps  a  train  at  constant 
speed  for  10  minutes,  how  much  heat  is  produced  in  the  rails  and 
bearings,  assuming  that  all  the  work  done  is  converted  into  heat? 

Ans.  6.65  X107  cals. 

42.  What  must  be  the  boiler  efficiency  in  order  that  a  Nordberg  quadruple 
expansion  engine  should  furnish  1  H.  P.  by  burning  1  Ib.  of  soft  coal 
per  hour?  Ans.  88  per  cent,  for  average  soft  coal. 

43.  Plot  the  Carnot  cycle  with  entropy  as  absciss®  and  thermodynamic 
temperature  as  ordinate.     What  does  an  area  on  this  diagram  repre- 
sent?    Derive  the  expression  for  the  efficiency  of  the  cycle. 


ELECTRICITY  AND  MAGNETISM 

BY  ALBERT  P.  CARMAN,  Sc.  D. 

Professor  of  Physics  in  the  University  of  Illinois 

MAGNETISM 

363.  Lodestones,  Magnets. — Pieces  of  iron-ore  are  sometimes 
found  which  show  a  strong  and  special  attraction  for  particles 
of  iron.     When  such  a  piece  of  iron-ore  is  dipped  into  iron  filings, 
the  filings  cling  to  it,  standing  out  in  tufts,  particularly  at  the 
edges  and  at  certain  points  on  the  piece  of  iron-ore.     A  piece  of 
iron-ore  which  shows  this  strong  and  special  attraction  for  iron 
is  called  a  lodestone  or  natural  magnet. 

By  methods  which  we  will  study  later,  a  piece  of  tempered 
steel,  such  as  a  knitting  needle  or  a  file,  can  be  magnetized,  that 
is,  can  be  made  to  acquire  the  same  property  as  the  lodestone 
for  attracting  iron.  A  "piece  of  magnetized  steel  is  often  called 
an  artificial  magnet  to  distinguish  it  from 
the  lodestone  or  natural  magnet.  It 
will  appear  later  that  there  are  no  es- 
sential differences  in  the  properties  of 
"artificial"  and  "natural"  magnets, 
and  we  shall  accordingly  use  the  term 
magnet  for  both  kinds.  Since  steel 
magnets  can  be  had  in  regular  and  con- 
venient forms,  they  are  better  adapted 
for  showing  the  properties  of  magnets 
and  will  be  used  altogether  in  our  study.  The  two  most  common 
shapes  given  to  such  magnets  are  the  U-shaped  or  horse-shoe 
magnet,  and  the  bar  magnet  (Fig.  228). 

Magnetism  is  a  term  used  for  the  science  of  magnets. 

364.  Magnetic  Poles. — When  a  magnet  is  dipped  into  iron 
filings,  it  is  seen  that  there  are  certain  points  or  regions  on  the 
magnet  of  maximum  attraction,  and  other  regions  where  the 
attraction  is  zero.     A  point  of  maximum  attraction  is  called  a 

309 


FIG.  228. 


310  ELECTRICITY  AND  MAGNETISM 

magnetic  pole,  and  a  region  of  no  attraction  is  called  a  neutral 
region.  It  is  found  that  a  magnet  always  has  at  least  two  poles. 
In  a  knitting  needle  as  commonly  or  " normally"  magnetized, 
there  are  two  poles,  one  near  each  end  of  the  needle,  while  the 
middle  of  the  needle  is  a  neutral  region.  This  distribution  is 
^  ^  shown  by  the  way  the  iron  filings 

*  cling  to  the  needle  (Fig.  229).    The 
parts  of  a  magnet  showing  magnetic 

attraction  are  said  to  have  free  polarity,  the  total  polarity  of 
the  middle  region  being  zero  (§§369,  375).  The  straight  line 
joining  the  two  magnetic  poles  is  called  the  axis  of  the  magnet. 

365.  Two  Kinds  of  Poles. — If  a  normally  ma-gnetized  knitting 
needle  is  suspended  so  that  it  rotates  freely  in  a  horizontal  plane 
about  its  middle  point,  the  needle  is  seen  to  come  to  rest  in  an 
approximately  north-and-south  line,  with  the  same  pole  always 
pointing  to  the  north,  and  with  the  other  pole  always  toward 
the  south.  The  magnetic  pole  which  points  northward  is  called 
the  north  (or  north-seeking)  pole,  and  the  other  pole  the  south 
(or  south-seeking)  pole.  The  north  pole  is  commonly  designated 
as  the  N  or  positive  (-f)  pole,  and  the- south  pole  as  the  S  or 
negative  pole. 

An  all  important  property  of  magnets  is  shown  by  bringing 
the  N  pole  of  a  magnet  near  the  N  pole  of  a  second  magnet  which 
is  suspended.  It  is  found  that  there  is  a  repulsion  between  these 
ends  of  the  two  magnets.  If  on  the  other  hand,-  the  N  pole  is 
brought  near  the  S  pole  of  the  suspended  magnet,  there  is  found 
to  be  an  attraction  between  the  two  magnets. 

Hence  we  have  the  law:  Like-named  magnetic  poles  repel  each 
other,  and  unlike-named  magnetic  poles  attract  each  other. 

The  above  law  gives  us  a  means  of  testing  whether  a  bar  of 
iron  or  steel  is  a  magnet  or  simply  a  magnetic  substance,  that 
is,  a  substance  attracted  by  a  magnet.  If  a  steel  bar  shows  at 
any  point  a  repulsion  for  the  N  pole  of  a  suspended  magnet,  the 
bar  is  a  magnet  and  the  repulsion  indicates  the  location  of  the 
N  pole  of  the  bar.  If  the  steel  or  iron  bar  is  not  magnetized, 
every  point  of  the  bar  shows  attraction  for  either  pole  of  the 
magnet. 

366.  Experiment  of  Breaking  a  Magnet. — When  a  strongly 
magnetized  needle  is  dipped  into  iron  filings  it  is  found  that  the 


MAGNETISM  311 

filings  cling  in  tufts  at  the  poles  near  the  ends,  but  that  there 

are  no  filings  near  the  middle,  or  the  neutral  region.     If  now  we 

break  the  needle  at  the  middle,  we  get  two  complete  magnets. 

Upon  testing  each  half,  we  find  that  a  S  pole  appears  on  the 

side  of  the  break  towards  the  original  N  pole,  and  a  N  pole  on 

the  side  towards  the  original  S  pole  (Fig.  230).     Each  half  can 

be  in  turn  broken,  and  four  mag- 

nets obtained  and  so  on  indefi-         \H  s\ 

nitely.     We  are  thus  lead  to  con- 

sider that  a  piece  of  iron  or  steel       ^ 

is  made  up  of  elementary  mag-     ^  -  ^  ^  -  m  w  -  ^  ^  -  ^ 

nets   with   equal   and   opposite 

Fid.  230. 

poles  and  that  the  magnetiza- 

tion of  a  bar  of  iron  or  steel  consists  in  arranging  these  elementary 
magnets  of  the  bar.  In  an  unmagnetized  steel  or  iron  bar,  there 
is  no  general  trend  of  these  elementary  magnets  in  any  one  direc- 
tion, and  so  they  neutralize  each  other's  external  action  (Fig. 
231);  if,  however,  we  can  by  any  means  turn  the  majority  of 
the  elementary  magnets  of  a  bar  in  the  same  general  direction, 
then  the  bar  becomes  a  magnet  (Fig.  232)  .  The  above  explana- 
tion of  the  action  of  a  magnet  has  been  termed  the  molecular 
theory  of  magnetism,  but  it  is  not  a  necessary  part  of  the  theory 
that  the  elementary  magnets  are  molecules  in  the  chemical 
sense  (see  §496  on  Electron  Theory  of  Magnetism). 


•3^*30  I    «3    §    (^  <fc 

Fio.  231.  Fio.  232. 

An  experiment  to  illustrate  the  above  theory  of  magnetism 
is  the  magnetization  of  steel  filings  contained  in  a  glass  tube. 
By  stroking  the  tube  on  a  strong  magnet  the  filings,  which  in 
general  are  magnets,  may  be  lined  up  in  the  direction  of  the  tube, 
so  that  there  results  a  N  pole  at  one  end  and  a  S  pole  at  the  other 
end.  This  is  shown  by  bringing  the  tube  up  to  a  delicately  sus- 
pended magnetic  needle.  If  now  the  filings  are  shaken  up,  so 
that  the  small  magnets  are  no  longer  lined  up  in  any  particular 
direction,  it  is  found  that  either  end  of  the  tube  of  filings  attracts 
either  pole  of  the  suspended  needle,  that  is,  the  tube  of  filings 
has  lost  its  magnetization. 


312 


ELECTRICITY  AND  MAGNETISM 


Swing's  model  of  a  magnet  consists  of  a  number  of  small 
magnetic  needles,  mounted  on  pivots  upon  a  glass  plate,  (Fig. 
233).  The  magnetic  needles  take  no  special  direction  unless 
acted  on  by  one^or  more  large  steel  magnets  or  by  the  directive 

magnetic  action  of  a  coil 
carrying  an  electric  cur- 
rent (see  §427).  Sw- 
ing's model  can  be  made 
of  a  size  to  be  put  in  the 
vertical  beam  of  a  pro- 
jection lantern,  and  thus  the  motions  and  directions  of  the  small 
magnets  can  be  clearly  shown  to  a  large  class.  This  model  is 
also  used  to  explain  properties  of  a  magnet  which  depend  upon 
the  mutual  action  of  the  elementary  magnets  (see  Hysteresis, 
§497). 

367.  Magnetic  Lines  of  Force,  Magnetic  Field. — We  have  seen 
that  a  magnet  acts  upon  a  neighboring  magnet,  the  unlike 


Fio.  233. 


FIG.  234a. 


poles  attracting  and  the  like  poles  repelling  each  other.  This 
action  was  assumed  by  earlier  writers  as  "  direct  action  at  a 
distance,"  that  is,  as  taking  place  across  space  and  not  by  means 
of  any  intermediate  actions.  Faraday  and  Maxwell  showed 
that  the  action  of  one  magnet  upon  another  is  to  be  explained 


MAGNETISM 


313 


as  due  to  lines  of  stress  which]  exist  in  the  space  between  the 
magnetic  poles.  These  lines  of  stress  can  be  traced  by  methods 
described  below,  and  are  called  "lines  of  magnetic  force." 


FlG.  2346. 


Fio.  234c. 


Magnetic  force  as  transmitted  along  these  lines  may  be  thought 
of  as  similar  to  a  "pull"  along  a  cord.    We  shall  see  later  that 


314 


ELECTRICITY  AND  MAGNETISM 


a  magnetic  line  of  force  is  probably  the  line  of  the  centers  of 
whirls  in  the  intervening  space.1 

The  lines  of  force  between  magnetic  poles  can  be  traced  by 
means  of  iron  filings.  Thus,  Fig.  234a  shows  the  tracings  made 
by  filings  of  the  magnetic  lines  between  a  N  and  a  S  pole.  It  is 
formed  by  sprinkling  iron  filings  on  a  glass  plate,  with  the  two 
poles  beneath  the  glass,  and  at  the  same  time  gently  tapping 


Fio. 

the  glass  so  that  the  filings  are  free  to  move.  The  filings  arrange 
themselves  in  lines  which  show  the  magnetic  lines  of  stress. 
Each  particle  in  the  filings  becomes,  for  the  time  being,  a  magnet, 
the  N  pole  of  which  tends  to  move  in  one  direction  along  a  line 
of  force,  while  the  S  pole  tends  to  move  in  the  opposite  direction. 
Fig.  2346  shows  the  lines  of  force  as  traced  by  lines  about  a  bar 
magnet.  In  Fig.  234c  we  have  the  lines  between  two  N  poles; 
in  Fig.  234d  the  lines  around  a  soft  iron  bar  in  the  field  between  a 
N  and  a  S  pole;  in  Fig.  234e  the  field  around  a  soft  iron  ring 
placed  between  a  A1  and  a  S  pole.  In  the  last  figure,  it  is  seen 
that  the  filings  form  no  lines  inside  of  the  ring,  that  is,  that 
region  is  shielded  from  the  magnetic  force  (see  §492) . 

1  Faraday  and  Maxwell,  and  moat  students  of  physics,  have  explained  the  transmission 
of  magnetic,  electrical  and  luminous  effects,  by  assuming  the  existence  of  a  medium  called 
the  "ether."  The  conception  of  the  "ether."  has  been  one  of  the  moat  helpful  and  con- 
venient theories  in  science,  but  it  has  never  been  without  difficulties.  To  explain  all  the 
observed  facts  of  magnetism,  electricity  and  light  requires  us  to  assume  a  medium  which 
has  "properties"  which  are  difficult  to  reconcile  with  each  other.  The  "ether"  is,  how- 
ever, the  best  working  hypothesis  which  we  have  for  these  phenomena  and  as  such  we 
use  it  in  the  discussion  of  electricity  and  magnetism. 


MAGNETISM  315 

368.  Magnetic  Field. — A  region  in  which  lines  of  magnetic 
force  exist  is  called  a  magnetic  field.  All  the  region  about  a  mag- 
net is  thus  a  magnetic  field.  It  will  be  shown  later  that  the  region 
about  an  electric  current  is  also  a  magnetic  field.  The  earth  is 
surrounded  by  a  magnetic  field,  known  as  the  earth's  mag- 
netic field.  A  sensitive  test  of  a  magnetic  field  is  the  exertion 
of  force  on  a  delicately  suspended  magnetic  needle.  Such  a 
magnetic  needle  is  acted  on  by  fields  which  are  too  weak  to  turn 
iron  filings.  Thus  the  earth's  field  does  not  rotate  iron  filings, 
but  it  acts  on  a  suspended  magnetic  needle. 

When  the  magnetic  lines  in  a  field  are  parallel  to  each  other, 
the  field  is  a  uniform  field.  The  earth's  magnetic  field,  in  places 


Fio.  234«. 

free  from  masses  of  magnetic  substances  and  distant  from  electric 
currents,  is  practically  uniform  over  considerable  areas.  A 
suspended  magnetic  needle  points  in  practically  the  same  direc- 
tion throughout  such  a  field. 

In  mapping  a  field  by  a  magnetic  needle,  we  note  that  the 
suspended  needle  places  itself  tangentially  to  the  magnetic  line 
through  its  center.  The  positive  direction  of  the  magnetic  line 
is  that  In  which  the  N  pole  of  the  needle  tends  to  move.  It  is 
thus  seen  that  a  magnetic  line  in  air  starts  from  the  N  pole  of  a 
magnet  and  ends  in  a  S  pole.  But  we  have  seen  that  a  N  and  a 


316  ELECTRICITY  AND  MAGNETISM 

S  pole  attract  each  other;  that  is,  the  magnetic  lines  tend  to 
shorten  or  contract.  Indeed,  in  Faraday's  thought,  the  attrac- 
tion between  the  two  unlike  poles  is  due  to  the  tension  of  the 
lines  which  join  the  poles,  these  magnetic  lines  acting  like 
stretched  rubber  cords.  It  is  noted  from  the  tracings  of  the 
lines  that  magnetic  lines  are  in  general  curves.  Faraday,  in  fact, 
often  referred  to  them  as  "magnetic  curves."  If,  however,  the 
only  property  of  a  magnetic  line  were  that  of  contraction,  the 
line  would  be  straight.  But  it  is  to  be  noted  that  lines  diverge 
from  each  other;  that  is,  the  general  form  of  the  lines  seems  to  be 
due  to  two  forces,  (a)  a  tension  along  the  line,  and  (6)  a  repulsion 
between  the  lines,  the  last  acting  like  a  pressure  at  right  angles 
to  the  lines.  James  Clerk  Maxwell  has  shown  mathematically 
that  the  properties  of  a  magnetic  field  and  the  resulting  forces 
acting  on  magnets  can  be  accounted  for  completely  by  the  longi- 
tudinal stress  and  the  lateral  or  perpendicular  pressure  in  the 
medium. 

369.  Methods  of  Magnetization. — We  can  magnetize  a  rod  of 
soft  Norway  iron  by  simply  holding  it  in  a  vertical  plane  through 
the  north-and-south  line  and  inclined  downward  about  70° 
from  the  horizontal;  after  a  very  little  tapping  or  perhaps  none, 
it  is  found  that  the  end  pointing  northward  has  become  a  N  pole. 
That  is,  the  elementary  magnets  of  the  rod  have  been  lined  up 
under  the  action  of  the  earth's  magnetic  field.  Upon  placing 
the  rod  at  right  angles  to  the  earth's  magnetic  field  and  again 
tapping  it  lightly,  it  is  found  to  have  lost  its  magnetization  as 
readily  as  it  gained  it.  The  rod  is  now  easily  magnetized  in  the 
opposite  direction  by  reversing  it  from  its  first  position,  and 
gently  tapping  it.  If  we  try  the  same  experiment  with  a  piece 
of  hard  iron  or  of  tool  steel,  it  is  found  that  the  hard  iron  or  steel 
can  be  magnetized  in  the  earth's  field  only  by  sharp  and  pro- 
longed tapping.  It  is  also  found  that  when  the  hard  iron  and  steel 
rods  are  once  magnetized,  they  retain  their  magnetization,  even 
when  their  position  in  the  field  is  changed.  This  property  of 
retaining  magnetization  is  called  magnetic  retentivity.  (The  term 
coercive  force,  once  used  for  magnetic  retentivity,  is  now  used  in  a 
different  sense.  See  §497  on  Hysteresis,  etc.).  The  elementary 
magnets  of  soft  iron  are  thus  easily  lined  up,  but  are  as  easily 
thrown  out  of  line  again.  The  elementary  magnets  of  steel 


MAGNETISM  317 

resist  a  change  of  direction,  and  hence  the  steel  is  less  easily 
magnetized,  but  when  once  it  is  magnetized,  it  retains  its  mag- 
netization.    Tool  steel  is  accordingly  adapted  for  permanent  mag- 
nets; soft  iron,  only  for  tem- 
porary magnets.     It  is  found 
that  the  retentivity  of  steel  is 
greatly  increased  by  temper- 
ing it,  so  that  strong  perma-  ^^  235* 
nent  magnets  are  always  made 
of  steel  tempered  hard. 
From  the  above  we  see  that 


the  process  of  magnetization  FIG.  236. 

consists  in  bringing  the  iron 

or  steel  into  a  magnetic  field.  Figs.  235  and  236  illustrate  what 
takes  place  according  to  the  molecular  theory.  In  the  first 
figure,  the  small  steel  magnets  point  in  all  directions,  and  the 
lines  of  force  are  practically  all  inside  the  group  of  magnets. 
In  the  second  figure,  the  small  magnets  are  pointed  in  one  gen- 
eral direction,  and  the  external  field  is  approximately  that  of 
a  bar  magnet. 

To  magnetize  an  iron  or  steel  bar  so  that  it  is  a  strong  magnet, 
it  is  necessary  to  line  up  a  large  part  of  the  elementary  magnets 
of  the  bar,  and  this  calls  for  a  strong  magnetic  field.  Hence, 
such  a  weak  magnetic  field  as  that  of  the  earth  gives  only  com- 
paratively small  magnetizing  effects.  Strong  magnetic  fields 
are  obtained  by  using  strong  steel  magnets,  or  strong  electro- 
magnets (§483)  or  solenoids  with  a  large  number  of  ampere- 
turns  (§430).  We  shall  study  later  (§484)  the  quantitative 
relations  between  the  strength  of  the  magnetic  field  and  resultant 
intensity  of  magnetization  for  various  kinds  of  iron  and  steel 
(§486). 

370.  Magnetic  Substances  and  Induced  Magnetism. — If  a  piece 
of  soft  iron  or  steel  such  as  a  nail  is  brought  near  the  N  pole  of 
a  strong  magnet,  not  only  is  it  attracted,  but  it  acquires  the 
property  of  attracting  other  nails;  thus  a  whole  series  or  chain 
of  nails  may  be  held  up  by  the  poles  "induced"  from  nail  to 
nail,  each  nail  becoming  for  the  time  a  magnet  (Fig.  237) .  That 
is,  the  attraction  is  really  an  attraction  between  the  N  pole  of 
the  magnet  and  the  S  pole  that  is  induced  in  the  nail.  Magnetic 


318  ELECTRICITY  AND  MAGNETISM 

substances  thus  are  substances  which  become  magnetic  by  in- 
duction and  hence  are  attracted  by  a  magnet.  Iron,  and  to  a 
less  degree,  nickel  and  cobalt  and  an  alloy  of  copper,  manganese 
^  and  aluminum,  called  the  "  Heus- 

1  ler  alloy,"  are  the  substances 
showing  magnetic  properties  most 
strongly,  and  are  called  ferromag- 
netic, but  many  other  substances 
show  a  slight  magnetic  attraction 

rlQ.  237.  .  _ 

m  very  strong  fields.  Such  sub- 
stances are  called  paramagnetic.  Still  other  substances  as  bis- 
muth, are  repelled  by  a  strong  magnet.  Such  substances  are 
called  diamagnetic.  The  quantitative  relations  of  the  magnetic 
properties  of  substances  will  be  discussed  under  Magnetic  Induc- 
tion (§485). 

371.  Intermediate  Poles. — As  "normally"  magnetized  a  needle  has  only 
two  magnetic  poles,  but  it  is  possible  to  magnetize  a  steel  needle  so  as  to 
have   more   than    two  points    of 

maximum  magnetic  attraction 
(Fig.  2386) .  In  Fig.  238a  is  shown 
one  method  of  securing  such  an  ir- 
regular magnetization.  The  bar 
is  placed  so  that  each  end  rests 

on  the  N  pole  of  a  bar  magne  Fl°*  238o> 

and  it  is  stroked  at  the  middle 
with  the  S  pole  of  a  third  magnet. 
The  bar  will  then  be  found  to  be 
magnetized  with  a  S  pole  at  each  Fio.  2386. 

end  and  a  N  pole  in  the  middle     It 

is  evident  that  we  have  in  this  case  the  equivalent  of  two  magnets  with 
the  two  N  poles  at  the  middle.  The  intermediate  pole  is  sometimes  called 
a  "consequent"  pole. 

372.  Coulomb's  Law  of  Magnetic  Force. — In  the  case  of  a 
slender  knitting  needle,  which  has  been  magnetized  in  a  strong 
uniform  magnetic  field,  the  elementary  magnets  are  generally 
so  perfectly  in  line  that  a  magnetic  pole  of  the  magnetized 
needle  can  be  assumed  for  distances  greater  than  a  few  centimeters 
to  be  a  point  very  near  the  end  of  the  needle.     (See  Fig.  229.) 
Coulomb,  a  French  physicist  and  mathematician,  in  1789  used 
slender  magnetic  needles  to  study  the  force  beween  two  magnetic 
poles  when  placed  at  different  distances.     He  found  that  the 


MAGNETISM 


319 


force  between  the  poles  varied  inversely  as  the  square  of  the 
distance  between  the  poles.  Coulomb's  method  of  experiment- 
ing with  "the  torsion  balance"  can  be  represented  diagram- 
matically  as  follows:  A  long  and  thin  needle  NS  (Fig.  239)  is 
suspended  horizontally  by  a  thin  silver  wire.  This  suspension 
wire  is  free  from  torsion  when  the  needle  is 
in  the  magnetic  meridian.  A  second  slender 
needle  N'S'  held  vertically  is  brought  so  that 
the  horizontal  distance  between  the  two  north 
poles  is  d  (as  measured  before  any  deflection  of 
NS  is  allowed).  If  NS  is  free  to  move,  it  is  de- 
flected by  the  repulsion  between  the  two  N  poles. 
To  bring  NS  to  its  original  position  a  twist  must 
be  given  to  the  suspension  wire,  by  turning  the 
torsion  head  until  the  force  of  torsion  is  equal 
to  the  force  of  repulsion  between  the  two  mag- 
netic poles.  The  force  between  the  two  poles 
is  measured  by  the  number  of  degrees  of  torsion 
in  the  wire  (§119).  By  thus  measuring  the 
forces  F',F",F'",etc.tfoT  the  distances  dlfd2,d3, 
etc.,  between  the  two  poles,  Coulomb  was  able 
to  show  that  F' :  F"  :  Fr"  ::  l/d\  :  l/d\  :  1/cP,, 
that  is,  that  F  is  proportional  to  l/da. 

We  can  now  define  the  c.  g.  s.  unit  magnetic  pole,  or  pole  of  unit 
strength;  a  unit  magnetic  pole  is  one  which  when  placed  at  one 
centimeter  distance  in  a  vacuum  from  an  equal  and  like  pole  repels 
it  with  a  force  of  one  dyne.  Hence,  if  the  centimeter  be  used  as 
the  unit  of  distance,  the  dyne  as  the  unit  of  force,  and  we  measure 
m  and  mf  in  terms  of  the  above  c.g.s.  unit  of  pole  strength, 
Coulomb's  law  for  a  vacuum  becomes 


If  instead  of  a  vacuum  there  is  a  material  intervening  medium,  a 
factor  for  that  medium  I//*  (§490)  must  be  used,  and  we  have 

F—  lmm' 

The  factor  1  ///  is  for  all  practical  purposes  equal  to  unity  for  air. 
The   proof  of  Coulomb's  law  does  not  rest   upon   Coulomb's 


320  ELECTRICITY  AND  MAGNETISM 

experiment,  which  is  necessarily  approximate,  but  upon  the  fact 
that  the  action  of  magnets  on  each  other  in  various  positions  can 
be  predicted  by  the  use  of  Coulomb's  law  (see  §377). 

373.  Intensity  or  Strength  of  a  Magnetic  Field. — The  force  F 
which  acts  on  a  magnetic  pole  placed  in  a  magnetic  field  depends 
upon  (a)  the  strength  m  of  the  pole  and  (&)  on  what  may  be 
called  the  strength  H  of  the  field.    This  suggests  the  following 
definition  of  the  strength  or  intensity  of  a  magnetic  field:     The 
strength  or  intensity  of  a  magnetic  field  at  a  point  is  equal  to -the 
number  of  dynes  of  force  which  act  on  a  unit  magnetic  pole  at 
the  point    Hence  F=mXH.    From  this  formula  we  can  calculate 
the  force  acting  on  a  magnetic  pole  if  we  know  the  pole  strength 
and  the  field  intensity,  it  being  assumed  that  the  strength  of 
the    magnetic   field  is  not  appreciably  changed  by  the  pres- 
ence of  the  testing  pole.     Thus  in  the  earth's  field  of  intensity 
0.6  a  pole  of  strength  4,  is  acted  on  with  a  force  of  0.6X4  =  2.4 
dynes. 

The  unit  field  intensity  is  by  some  writers  called  the  gauss. 
Thus  the  earth's  magnetic  field  at  Washington  would  be  de- 
scribed as  a  field  of  0.6  gausses. 

374.  Quantitative  Use  of  Lines  of  Force. — Magnetic  lines  of 
force,  as  they  have  been  defined  above  (§367),  fix  only  the  direc- 
tion of  the  field.     The  fact  that  in  the  figures  made  by  iron  filings, 

the  lines  appear  most  numerous  where  the 
field  is  strongest,  suggests  that  the  intensity 
of  the  field  may  be  represented  by  the  num- 
ber of  the  lines  of  force.  To  this  end,  we 
agree  to  restrict  the  number  of  lines  drawn 
to  represent  a  magnetic  field  so  that  in  a  field 
of  unit  intensity  there  is  one  line  of  force 
240  Per  scluare  centimeter  of  normal  section,  and 

in  a  field  of  intensity  H  there  are  H  lines 
per  square  centimeter;  or  the  intensity  of  the  field,  as  defined 
above,  is  numerically  equal  to  the  number  of  lines  per  square  cen- 
timeter cutting  a  plane  at  right  .angles  to  the  field.  That  a  line 
is  continuous  in  the  field  will  appear  from  a  consideration  of 
the  lines  entering  and  leaving  poles,  and  also  from  the  very 
nature  of  a  line  of  force. 

If  the  field  is  uniform,  the  total  number  of  lines  across  the 


MAGNETISM 


321 


field  of  section  S  would  thus  be  N—SH.     If  the  field  is  not  of 
uniform  intensity, 

N  =  (SlHl  +  £3#3  + ,  etc.)  =  ISH 

By  the  following  consideration  we  see  that  4nm  lines  of  force 
emerge  from  a  pole  +m  in  a  vacuum.  Describe  about  the  pole 
m  as  center  a  spherical  surface  with  radius  r  (Fig.  240).  The 
intensity  of  the  field  on  the  sphere  is  by  Coulomb's  law 

,r      TttXl 


This  field  H  is  evidently  at  right  angles  to  the  surface  of  the  sphere. 
Hence  there  are  ra/r2  lines  across  each  square  centimeter  of  area 
of  the  sphere.  The  total  number  of  lines  coming  from  the  pole 
is  therefore 


-mH 


X' 


Since  this  is  true  for  any  and  every  value  of  r,  these  lines  are 
continuous.     In  a  similar  way  we  find   that 
47rw  lines  enter  the  pole  —  ra. 

375.  Forces  on  a  Magnet  in  a  Magnetic 
Field.  —  A  magnetic  needle  in  a  uniform  mag- 
netic field,  such  as  the  earth's  field,  is  acted 
on  by  two  equal  and  opposite  forces,  the  force 
+  mH  on  the  N  pole,  and  the  force  —mH  on 
the  S  pole,  Fig.  241.  We  thus  have  two  equal 
and  opposite  forces  acting  at  opposite  ends 
of  the  magnet,  that  is,  we  have  a  couple  (§98). 
The  action  on  the  magnetic  needle  is  simply 
to  rotate  it  into  the  line  of  the  field,  without 
translation.  This  can  be  easily  verified  by 
floating  a  magnetic  needle  on  a  cork  in  a  large 
basin  of  water.  The  needle  is  not  drawn  to 
the  north  nor  to  the  south  but  simply  ro- 
tates and  finally  comes  to  rest  in  the  magnetic  meridian. 
can  also  regard  the  experimental  fact  that  there  is  no  tractive 
force  on  a  magnet  in  the  uniform  field  of  the  earth,  as  a  proof  of 
the  assumption  made  above,  that  the  two  poles  +m  and  —  m,  are 
equal  in  strength. 

If  the  field  is  not  uniform,  but  is  stronger  at  one  pole  than  at 
the  other,  there  will  be  a  tractive  force  on  the  magnet.     It  is 
21 


I 


Fia.  241. 


We 


322  ELECTRICITY  AND  MAGNETISM 

then  not  only  rotated,  but  also  acted  on  by  a  resultant  force 
in  one  direction  or  the  other. 

376.  Torque  in  a  Uniform  Field. — The  torque  or  moment  of 
the  couple  acting  on  a  magnetic  needle  in  a  uniform  field  is  easily 
expressed.  Consider  the  magnet  NS  of  length  Z,  making  the 
angle  0  with  the  direction  of  the  field  XX'  (Fig.  241).  If  +m 
and  —  m  are  the  strengths  of  the  poles,  and  H  the  intensity  of 
the  field,  the  field  exerts  two  parallel  forces  +  mH  and  —mH,  and 
the  moment  of  the  couple  is  L=mHx  (arm  of  the  couple).  The 
arm  of  the  couple  =2Np  =  20N  sin  0—1  sin  0. 
Hence  L  =  Hml  sin  6. 

Put  ml  =  M,  then  L  =  HM  sin  6. 

The  term  ml  is  called  the  magnetic  moment  of  the  magnet.  The 
magnetic  moment  of  a  magnet  is  equal  to  the  product  of  the  pole 
strength  of  the  magnet  by  the  distance  between  the  poles.  When 
the  magnet  is  held  at  right  angles  to  the  field,  that  is,  when 
19  =  90°,  the  torque  is  L  =  HM. 

If  the  field  strength  H  is  unity,  then  the  torque  L  is 
equal  to  the  magnetic  moment  M,  and  it  follows  that:  The 

magnetic  moment  of  a  magnet 

%1r~ — ^rr  ~^\sl  1S    numerically    equal    to    the 

NJ^^^  ^^s8*   t°r(lue  Acting  on  it  when  it  is 

242  held    at  right  angles   to   unit 

field. 

It  is  to  be  noted  that- the  magnetic  moment  of  a  magnet  is  a 
quantity  that  admits  of  exact  determination,  while  the  strength 
of  the  pole  m,  and  the  distance  I  between  the  two  poles,  cannot 
be  exactly  determined  in  a  physical  magnet. 

It  is  easy  to  see  that  the  magnetic  moment  of  a  bundle  of 
magnets  is  equal  to  the  algebraic  sum  of  the  magnetic  moments 
of  the  individual  magnets.  But  a  physical  magnet  is  to  be 
looked  upon  as  a  bundle  of  magnetic  filaments.  By  a  magnetic 
filament  we  mean  a  single  line  of  elementary  magnets,  arranged 
as  in  Fig.  242,  with  only  two  poles  "free,"  that  is,  poles  not 
neutralized  by  the  presence  of  equal  opposite  poles.  These  two 
"free"  poles  N  and  /S,  form  the  poles  of  the  filament.  Hence  the 
magnetic  moment  of  the  whole  magnet  is  the  algebraic  sum  of 
the  magnetic  moments  of  these  filaments  of  the  elementary 
magnets. 


MAGNETISM  323 

377.  Calculation  of  the  Intensity  of  the  Magnetic  Field  in 
Special  Cases. — It  is  possible  by  the  use  of  Coulomb's  law  to 
calculate  the  strength  of  the  magnetic  field  at  certain  points 
about  a  magnet  of  known  magnetic  moment.  The  cases  of  most 
importance  are  for  the  two  positions  known  as  "position  A" 
and  "position  B  of  Gauss." 

For  Position  "  A ." — Consider  the  strength  of  the  magnetic  field, 
due  to  a  bar  magnet  at  a  "distant"  point  on  the  line  of  its  axis. 
The  strength  of  the  pole  is  m,  the  distance  between  the  poles 
or  length  of  the  magnet  is  2L;  the  problem  is  to  find  the  strength 
of  the  field  at  a  point  P  in  the  line  of  the  axis,  and  distant  r  from 
the  mid-point  of  the  axis  of  the  magnet  (Fig.  243).  By  Cou- 
lomb's law,  the  force  on  a  unit  positive  pole  at  P  is 


+m 

Fio.   243. 


_jn  m_ 

(r-L)'     (r+L) 


If  P  is  "distant,"  so  that  L2  can  be  neglected  as  compared  with 
ra,  we  get 


or  the  strength  of  field  at  P  is 

ffj,_2£.  (A) 

The  direction  of  this  field  is  evidently  that  of  the  axial  line  OP. 

For  Position  "B."  —  Consider  the  strength  of  field  due  to  a 
magnet  at  a  "distant"  point  on  the  line  bisecting  the  axis  at 
right  angles  (Fig.  244). 

In  this  case  the  forces  acting  on  unit  pole  at  P,  are  a  repulsion 
due  to  +m,  represented  by  PA,  and  an  attraction  due  to  —  m, 
represented  by  PB.  The  resultant  is  represented  by  the  diag- 


324 


ELECTRICITY  AND  MAGNETISM 


onal  PR.     Since  the  triangles  PAR  and  NPS  are  similar,  we 
have, 

PR     NS          2L 


But  PA  represents  the  force  exerted  by  -f  m  on  the  unit  posi- 
tive pole  at  P,  or  from  the  law  of  Coulomb 


PA  = 


m 


(r»+L»)f 


-m 


2L 


+  m 


Fro.  244. 


substituting   this   value  for   PA,  and   transposing   we  get  the 
resultant  force 


If  P  is  "distant/'  so  that  La  can  be  neglected  as  compared  with 
ra,  we  get 


The  direction  of  this  field  is  evidently  perpendicular  to  the  bi- 
secting line  OP,  or  parallel  to  the  magnet. 

From  the  above  it  is  seen  that  the  intensity  of  the  field  for 
position  "  A  "  is  twice  that  for  position  "  B"  for  the  same  magnet 
and  the  same  distance.  As  the  calculations  have  been  made  on 
the  assumption  of  Coulomb's  law,  we  have  here  a  means  of  testing 
this  law  by  comparing  experimentally  the  intensities  of  the  fields 
in  the  two  cases.  This  has  been  done  by  Gauss  and  the  results 
of  experiments  agree  with  the  law. 

378.  Methods  of  Comparing  the  Intensities  of  Two  Fields.  — 
Since  the  force  acting  on  a  magnetic  pole  is  proportional  to  the 
intensity  of  the  field,  (that  is,  F=mH)  the  ratio  of  the  intensities 


MAGNETISM  325 

of  two  fields  is  equal  to  the  ratios  of  the  forces  which  act  on  the 
same  magnetic  pole  in  the  two  fields;  thus  H1  :  Ht  ::  Fl  :  F2, 
where  Hl  and  H^  are  the  intensities  of  the  two  fields,  and  Ft  and 
FI  are  the  two  forces  on  the  pole  m  in  these  fields.  The 
forces  Fj  and  F2  can  be  measured  by  the  following  methods: 
(a)  By  balancing  the  torque  on  a  suspended  magnet  by  the  tor- 

sion of  a  suspension  wire. 
(6)   By  the  vibrations  of  an  oscillating  magnet. 
(c)    By  the  deflections  produced  by  a  second  magnet. 

379.  Comparison  of  Magnetic  Fields  by  the  Torsion  Balance.  —  First 
suspend  the  magnet  by  a  wire  (or  quartz  fiber)  suspension,  and  arrange 
so  that  there  is  no  torsion  in  the  suspension  wire  when  the  needle  is  in 
the  direction  of  the  field.  Next  twist  the  wire  by  means  of  the  "torsion 
head"  until  the  needle  is  deflected  through  a  given  angle  <f>  The  number 
of  degrees  of  torsion,  x,  in  the  wire,  is  calculated  from  the  reading  of  the 
torsion  head  and  the  deflection  of  the  magnet.  Then  xl^kMHl  sin  <pl 
(§376)  where  k  is  a  constant  for  a  given  suspending  wire  (§168).  If  we 
repeat  this  experiment  with  the  same  magnet  and  the  same  suspension 
arrangements  in  a  second  magnetic  field  we  find  a  torsion  xt  for  deflected  $,. 
Thus  s2  *=kMH3  sin  <£a.  From  this  it  follows  directly  that 


If  the  deflection  <£x  be  made  equal  to  <f>v  the  proportion  becomes 


380.  Comparison  of  Fields  by  the  Oscillations  of  a  Magnet.  — 

When  a  suspended  magnet  is  deflected  through  an  angle  0  from 
the  direction  of  the  field,  it  is  acted  on  by  a  restoring  couple 
MH  sin  6  (§376).  For  small  angles,  the  sine  and  the  angle  are 
assumed  equal,  and  hence  the  restoring  couple  is  proportional 
to  0,  and  MH  6—  —  7a,  where  /  is  the  moment  of  inertia,  and  a 
the  angujar  acceleration  (see  §89).  Hence  the  motion  agrees 
with  the  definition  of  angular  harmonic  motion  (§118),  and  the 
period  T  is  given  by  the  formula 


Transposing  we  get, 


M 


where  n  is  the  frequency  of  the  vibration.     By  allowing  the  same 
needle  to  vibrate  in  two  fields  of  strength  Hl  and  H2,  and  noting 


326 


ELECTRICITY  AND  MAGNETISM 


the  periods  Tl  and  T7,  or  frequencies  nl  and  n2;  we  get  the  pro- 
portion, 

H,:  #2 


Fio.  245. 


In  the  above,  it  is  assumed  that  the  moment  of  the  magnet  is 
constant,  and,  therefore,  the  method  cannot  be  used  in  strong 
fields,  that  is,  in  fields  which  change  the 
moment  of  the  magnet  by  induction. 
Fig.  245  shows  simple  apparatus  for 
making  observations  of  the  oscillations 
of  a  magnet.  The  magnet  is  a  cylinder, 
the  moment  of  inertia  of  which  can  be 
calculated  by  formula  (§93). 

381.  The  Tangent  Law. — When  a  mag- 
net is  under  the  action  of  two  fields  H 
and  R,  which  are  at  right  angles  to  each 
other,  it  takes  a  resultant  position  mak- 
ing an  angle  0  with  H,  and  an  angle 
(90°-0)  with  R,  (Fig.  246).  The  moment  of  the  couple  tending 
to  rotate  it  into  the  direction  of  H  is  L,  =  M7/  sin  6,  and  that 
into  the  direction  of  R  is  L^MR  sin  (90°-0)  =MR  cos  6. 
Since  the  magnet  comes  to  rest  at  the  deflection  6,  the  two 
opposite  torques  Ll  and  L2  must  be  numerically  equal,  that  is, 
L1  =  Z/2  or 

MH  sin  0  =  MR  cos  0 
From  this  we  get 

R     am  0 
--——  --  «=tan  0 
H    cos0 

Hence:  //  a  magnetic  needle  in  a  field  of  intensity  H  is  deflected 
through  an  angle  0  by  a  field  R  at  right  angles  to  H,  then  the  tangent 
of  the  angle  of  deflection  6  is  equal  to  the  ratio  of  the  strengths  of 
the  two  fields  R  and  H. 

The  tangent  law  is  used  in  the  tangent  galvanometer  (§436), 
and  most  other  magnetic  deflection  instruments.  It  is  an  applica- 
tion of  the  general  law  that  the  ratio  of  two  rectangular  forces 
is  equal  to  the  tangent  of  the  angle  which  the  resultant  makes 
with  the  first  component. 

382.  Comparison  of  Magnetic  Fields  by  the  Deflection  Experi- 
ment.— In  this  method  a  small  magnetic  needle  is  deflected  from 


MAGNETISM 


327 


the  direction  of  the  field  by  a  second  magnet,  which  is  placed 
so  as  to  produce  a  field  at  right  angles  to  the  field  to  be 
measured.  A  simple  form  of  apparatus  for  this  experiment  is 
shown  in  Fig.  247  a  and  b.  It  consists  of  a  magnetic  compass  0 


Fio.  246. 


mounted  in  the  middle  of  a  graduated  bar  A  B.  The  compass 
box  is  arranged  with  graduated  circle  so  that  the  deflection  of  the 
needle  can  be  read.  The  bar  AB  is  set  at  right  angles  to  the 
magnetic  field  Hlt  and  the  zero  position  of  the  needle  is  read  on 
the  graduated  circle.  A  magnet  NS,  is  now  placed  at  a  point 


Fio.  247a. 


B  on  the  bar,  and  it  then  produces  a  magnetic  field  of  strength  R 
at  right  angles  to  the  field  Hr  The  needle  takes  a  resultant 
position,  making  an  angle  0^  with  the  field  Hlt  such  that 
72/.flr1  =  tan  0t  (see  §381).  We  can  now  transfer  the  apparatus 
into  a  second  magnetic  field  Hz,  and  get  a  second  angle  of  de- 


328 


ELECTRICITY  AND  MAGNETISM 


flection  0r     Then  R/HZ=  tan  02.     Dividing  the  second  equation 
by  the  first,  we  get, 


The  ratio  of  the  tangents  of  the  two  angles  of  deflection  thus  gives 
the  ratio  of  the  intensities  of  the  two  magnetic  fields. 

0 


N 


FIG.  2476. 

383.  Determination  of  H  and  M  in  Absolute  Measurements. — 
The  methods  described  above  (§§380,  382)  are  comparison 
methods,  that  is,  the  relative  strengths  of  magnetic  fields  are 
determined,  but  not  their  absolute  values.  The  absolute  meas- 
urements of  a  magnetic  field  such  as  that  of  the  earth  can  be 
made  by  a  combination  of  the  oscillation  and  the  deflection 
experiments  as  follows: 

First,  a  magnet  of  moment  M  is  allowed  to  vibrate  freely  in  a 
field  of  strength  H,  and  we  thus  get  the  relation  (§380), 

HM-~^~  (I) 

By  this,  the  value  of  the  product  HM  is  obtained. 

Second,  using  the  same  magnet  of  moment  M  as  the  deflect- 
ing magnet  in  the  deflection  experiment,  (§381),  we  get 

e 


But  in  §377,  we  have  seen  that  the  field  due  to  a  magnet  in  the 
"A"  position  is  R  =  2M/r*.  Substituting  this  value,  we  get 

?-„ ?_  (II) 

M    r8tan0 

where  r  is  the  distance  in  centimeters  from  the  center  of  the 
magnet  M  to  the  center  of  the  deflected  magnet  (Fig.  247) ,  0  is  the 
angle  that  the  deflected  magnet  makes  with  the  field  H,  and  M 
is  the  magnetic  moment  of  the  deflecting  magnet.  Combining 
equation  (I)  and  (II),  we  can  eliminate  M,  and  get  H  in  terms 
of  observed  quantities  and  numbers,  that  is,  in  absolute  measure. 


MAGNETISM  329 

(This  "strength  of  field"  is  the  horizontal  component  of  the  earth's 
field,  see  §387.) 

In  a  similar  way  we  can  get  M ,  the  magnetic  moment  of  the 
magnet,  in  absolute  measure  by  eliminating  H  between  the  equa- 
tions (I)  and  (II). 

384.  Magnetometers  to  Determine  the  Horizontal  Component 
of  the  Earth's  Magnetic  Field. — By  the  use  of  simple  apparatus 
such  as  shown  in  Figs.  245  and  247,  the  value  of  H,  the  horizontal 
component  of  the  earth's  magnetic  field,  can  be  determined  to 
an  accuracy  of  a  few  per  cent.     For  the  most  accurate  work, 
such  as  is  required  in  the  magnetic  surveys  of  the  governments 
of  the  United  States  and  Great  Britain,   the  "Kew"  unifilar 
magnetometer  is   used.     This  is  shown  in  Fig.  248   arranged 
for   deflection    experiments.     The  general  method  is  that   of 
the  simpler  apparatus,  but  special  details  and  corrections  are 
involved   for   which    the    larger   laboratory  manuals  must  be 
consulted. 

385.  The  Earth  a  Magnet.— The  fact  that  a  suspended  mag- 
netic needle  tends  to  place  itself  in  a  north-and-south  line,  led 
to  the  theory  that  "the  globe  of  the  earth  is  a  great  lodestone," 
and   that  the  positive  magnetic  pole  of   the  earth  is  near   its 
south  geographical   pole,   and    its  negative   magnetic    pole  is 
near  its  geographical  north  pole.     Sir  William  Gilbert,  rightly 
called  "the  father"  of  magnetism  as  a  science,  first  published 
this  theory  in  1600  in  his  famous  book  the  "  De  Magnete."     But 
later  study  has  shown  that  the  magnetization  of  the  earth  is 
very  complex  and  the  two  so-called  "magnetic  poles"  of  the 
earth,  in  the  northern  and  southern  hemispheres  respectively, 
must  not  be  regarded  as  closely  analogous  to  the  pole  of  an 
ordinary  magnet,  but  are  merely  places  where  the  magnetic 
force  is  perpendicular  to  the  earth's  surface. 

The  study  of  the  earth's  magnetic  field  is  one  of  the  most  im- 
portant and  interesting  fields  of  science,  because  it  involves  the 
problem  of  how  and  why  the  earth  is  a  magnet,  and  also  because 
of  the  use  of  the  magnetic  compass  in  navigation  and  surveying 
and  in  the  absolute  electrical  measurements.  A  complete  descrip- 
tion of  the  earth's  magnetic  field  calls  for  determinations  of  (a) 
the  direction,  and  (b)  the  intensity  of  the  field  for  every  part  %of 
the  earth. 


330 


ELECTRICITY  AND  MAGNETISM 


386.  Direction  of  the  Earth's  Magnetic  Field.     Declination,  Dip. 

—It  is  found, that  a  magnetic  needle  which  is  suspended  so  as 
to  rotate  in  a  horizontal  plane,  does  not  in  general  point  exactly 
to  the  geographical  north.  The  angle  which  such  a  needle  makes 
with  the  geographical  meridian  is  called  the  declination.  This 
angle  varies  with  both  place  and  time.  Thus,  in  1905,  the  dec- 
lination of  the  magnetic  needle  at  London  was  16°  32.9'  W  of 


Fio.  248. 

north;  at  New  York  it  was  9°  0.8'  W,  and  at  San  Francisco  16° 
55'  E.  There  are  also  variations  with  time,  but  these  are  generally 
slow  or  transient  and  will  be  considered  in  §389.  A  vertical 
plane  through  the  axis  of  a  compass  needle  intersects  the  earth 
in  a  line  called  the  magnetic  meridian.  Evidently  the  declina- 
tion at  any  place  can  be  defined  as  the  angle  between  the 
geographical  and  magnetic  meridians. 

About  1544,  Hartmann  observed  that  a 
needle  which  was  balanced  horizontally  when 
non-magnetized,  was  no  longer  balanced  when 
magnetized,  but  dipped  with  its  north  pole 
downward.  The  magnetic  dip  or  inclination 
thus  observed  can  be  measured  by  an  instru- 
ment called  a  dip  circle.  This  consists  (Fig. 
249)  of  a  vertical  graduated  circle  which  can 
be  set  in  the  magnetic  meridian.  At  the 
center  of  the  circle  is  a  magnetic  needle 
which  is  balanced  on  a  horizontal  axis  through 
its  center  of  gravity,  so  as  to  rotate  freely 


Fio.  240. 


MAGNETISM  331 

in  the  plane  of  the  magnetic  meridian.  The  angle  which  a  mag- 
netic needle  balanced  at  its  center  of  gravity  makes  with  the 
horizontal  is  called  the  dip  or  inclination.  In  the  northern  hemi- 
sphere, the  north  pole  of  the  needle  dips  downward,  or  the  dip 
is  positive,  while  in  the  southern  hemisphere,  the  south  pole  of 
the  needle  dips  downward,  or  the  dip  is  negative.  The  line  of 
no  dip,  which  encircles  the  earth  near  the  equa- 
tor, is  called  the  magnetic  equator.  When  the 
declination  and  the  dip  are  known,  the  direction 
of  the  magnetic  field  is  evidently  determined. 

387.  Intensity  of  the  Earth's  Magnetic  Field. 
— The  method  of  §383,  for  the  determination  of 
the  intensity  of  a  magnetic  field  by  using  the 
same  magnet  in  deflection  and  oscillation  experi- 
ments, applies  to  the  earth's  field.  Evidently  the 


intensity  thus  determined  is  that  in  the  horizon- 
tal direction,  or  the  horizontal  component  H  of  the  earth's  field. 
If  we  know  the  dip  <f>,  and  the  horizontal  component  H,  we 
get  directly  the  total  intensity,  T,  that  is  (Fig.  250), 


The  vertical  component  V  is  also  given  by  the  relation 


388.  Magnetic  Maps.  —  The  results  of  the  magnetic  surveys, 
in  which  the  declination,  the  dip  and  the  intensity  of  the  earth's 
magnetic  field  at  various  places  have  been  determined,  are  best 
shown  by  means  of  lines  drawn  on  a  map.  A  line  drawn  through 
points  having  the  same  declination  is  called  an  isogonic  line 
Fig.  251  shows  the  isogonic  and  the  agonic  lines  for  the  world. 
The  amount  of  the  declination  is  indicated  by  the  figures  on 
the  line.  Thus  the  line  passing  near  New  York  is  that  of  10°  W. 
declination.  It  is  seen  that  the  line,  passing  near  Cincinnati, 
Ohio,  has  a  declination  of  0°.  This  is  an  agonic  line. 

A  line  connecting  points  having  the  same  dip  or  inclination 
is  called  an  isoclinic  line.  The  isoclinics  follow  the  general 
direction  of  the  parallels  of  latitude,  and  some  of  them  are  indi- 
cated by  dotted  lines  in  Fig.  251. 

Over  a  magnetic  pole,  the  dip  is  90°.  A  magnetic  pole  is  not 
at  the  corresponding  geographical  pole.  The  one  in  the  northern 


332 


ELECTRICITY  AND  MAGNETISM 


MAGNETISM 


333 


hemisphere  is  at  present  in  the  neighborhood  of  97°  W.  long., 
and  75°  N.  lat.;  but  the  magnetic  poles  of  the  earth  are  not  to  be 
thought  of  as  definite  points. 

A  line  connecting  points  of  the  same  intensity  is  called  an 
isodynamic  line. 

389.  Time  Variations  of  the  Declination. — Observations  of  the 
magnetic  decimation  have  been  taken  in  western  Europe  with 
more  or  less  regularity  since  1580.  These  observations  show  that 
in  1580  there  was  an  easterly  declination  in  London  of  11°  15' 
which  decreased  until  it  was  zero  in  1657,  and  then  became 


Sun 


spots. 


Magnetic 
variations 


FIG.  252. 


westerly,  reaching  a  maximum  westerly  decimation  of  about 
24°  38'  in  1818;  since  that  time  it  has  been  decreasing.  In  recent 
years  the  decimation  has  been  decreasing  at  about  5'  per  year. 
This  variation  is  called  the  secular  variation  of  the  declination. 
Observations  also  show  similar  secular  variations  of  inclination 
and  intensity.  In  addition  to  the  above  there  are  variations  of 
the  earth's  magnetic  elements,  which  have  annual  and  daily 
periods.  A  very  interesting  fact  in  terrestrial  magnetism  is  that 
times  of  disturbances  on  the  surface  of  the  sun  are  times  of 
maximum  changes  in  the  earth's  magnetism.  Thus  Fig.  252 
given  by  Bigelow  in  the  U.  S.  Monthly  Weather  Review,  shows 


334  ELECTRICITY  AND  MAGNETISM 

that  the  eleven-year  period  of  the  sun  spots  corresponds  to 
periods  of  maximum  magnetic  variations.  The  phenomena  of 
the  aurora  borealis  are  also  closely  connected  with  magnetic  dis- 
turbances. 

Why  the  earth  is  a  magnetized  body  has  been  a  much  debated 
question.  Among  the  causes  discussed  have  been,  distributions 
of  magnetized  masses  in  the  earth,  and  the  presence  of  electric 
currents  in  the  earth  and  in  the  atmosphere.  No  complete  and 
satisfactory  theory  has,  however,  been  reached.  For  more  ex- 
tended discussions  of  terrestrial  magnetism,  the  student  is  referred 
to  the  article  in  the  eleventh  edition  of  the  Encyclopaedia  Britan- 
nica  and  to  various  articles  in  the  journal,  Terrestrial  Magnetism. 

ELECTROSTATICS 

390.  Fundamental  Experiments. — If  a  rod  of  hard  rubber  is 
rubbed  with  fur,  it  is  found  that  light  particles  are  attracted 
to  the  rod.     Thus  shreds  of  paper  and  pieces  of  pith  cling  to  the 
rubber,  and  after  a  short  contact  are  strongly  repelled.     A  very 
convenient  instrument  for  detecting  this  attraction  and  repulsion 
is  a  small  gilded  pith  ball  hung  by  a  silk  fiber.     A  body  which 
has  acquired  this  property  of  attracting  and  then  repelling  light 
particles  is  said  to  be  electrified.     The  cause  of  this  attraction  is 
ascribed  to  an  agent  called  "electricity,"  and  the  electrified 
body  is  said  to  have  "a  charge  of  electricity,"  or  simply  to  be 
"charged."     A  suspended  pith  ball  or  other  device  for  detecting 
electrification  is  called  an  electroscope. 

This  electrified  state  may  be  acquired  similarly  by  other 
substances.  Among  the  substances  which  show  it  very  strongly 
are  amber,  rubber,  resin,  sulphur,  sealing  wax  and  shellac  when 
rubbed  with  fur,  and  glass  and  crystals  when  rubbed  with  silk. 
It  will  be  seen  later  that  electrification  results  when  any  two 
different  substances  are  rubbed  together,  but  that,  in  most  cases, 
it  can  be  detected  only  by  special  appliances. 

391.  Two  Kinds  of  Electrification. — If  a  rubber  rod  which  has 
been  electrified  by  friction  with  fur  be  suspended  by  a  thread 
so  that  it  is  free  to  rotate  in  a  horizontal  plane  about  its  middle 
point,  it  is  found  that  this  rod  is  repelled  by  a  similarly  electrified 
rubber  rod.     If  next  a  glass  rod  be  electrified  by  friction  with  silk, 


ELECTROSTATICS  335 

and  brought  near  the  suspended  electrified  rubber  rod,  there  is  an 
attraction.  In  the  same  way,  the  electrified  glass  rod  can  be 
suspended,  and  it  is  found  that  it  is  repelled  by  another  electrified 
glass  rod.  That  is,  the  electrification  of  glass  from  friction  with 
silk  acts  in  an  opposite  way  to  the  electrification  of  rubber  from 
friction  with  fur;  or,  in  other  words,  there  are  two  kinds  of 
electricity.  The  electricity  on  the  glass  is  called  positive  electric- 
ity; while  that  on  the  rubber  is  called  negative  electricity.  The 
above  experiments  show  that  bodies  charged  with  like  kinds  of 
electricity  repel  each  other,  and  bodies  charged  with  unlike  kinds  of 
electricity  attract  each  other. 

We  have  seen  that  a  rubber  rod  becomes  electrified  negatively 
by  friction  with  fur.  If  we  now  test  the  fur,  we  find  that  it  also 
is  electrified,  but  that  it  attracts  the  electrified  rubber,  and  repels 
the  electrified  glass  rod.  That  is,  the  fur  becomes  electrified 
positively  at  the  same  time  that  the  rubber  becomes  electrified 
negatively.  In  the  same  way,  experiment  shows  that  in  the 
friction  of  glass  and  silk,  the  silk  becomes  electrified  negatively, 
while  the  glass  is  being  electrified  positively.  In  general,  when 
electrification  is  produced  by  the  friction  of  two  different  sub- 
stances, both  substances  are  electrified,  the  one  with  one  kind  of 
electrification  and  the  other  with  the  opposite  kind  of  electrifica- 
tion. In  the  following  list,  a  number  of  common  substances  are 
arranged  in  a  so-called  "  electric  series, "  the  order  being  chosen 
so  that  if  a  substance  be  rubbed  with  a  second  substance  which 
is  further  down  the  series,  the  first  substance  becomes  positively, 
and  the  second  substance  negatively  electrified.  Thus  when 
glass  is  rubbed  with  silk,  the  glass  is  positive  and  the  silk  negative, 
while  glass  rubbed  with  fur  becomes  negatively  electrified.  The 
electrification  of  a  substance,  however,  depends  so  largely  upon 
the  surface  conditions,  impurities,  temperature,  etc.,  that  the  order 
in  the  series  is  only  approximate. 

fur  glass         metals  resin 

wool  silk          hard  rubber         sulphur 

quartz  wood        sealing  wax          gun  cotton 

392.  Transfer  of  Electricity,  Conductors  and  Insulators. — If  all 
points  of  an  electrified  rod  be  touched  to  a  metal  ball  which  is 
held  in  the  hand,  it  is  found  that  the  rubber  has  lost  all  of  its 


336  ELECTRICITY  AND  MAGNETISM 

electrification;  and  further,  if  the  experiment  is  repeated,  ex- 
cept that  the  ball  is  mounted  on  a  glass  or  rubber  stand,  it  is  found 
that  the  ball  has  acquired  the  electrification  of  the  rubber,  and 
that  it  attracts  and  then  repels  light  particles,  such  as  the  sus- 
pended pith  ball.  By  touching  the  mounted  ball  to  a  second 
mounted  ball,  a  portion  of  the  electrification  is  again  transferred. 
Electrification  can  thus  be  transferred  from  body  to  body  by 
conduction.  The  repulsion  of  the  pith  ball  after  it  has  touched  a 
charged  body,  is  due  to  its  becoming  charged  by  conduction  with 
the  same  kind  of  electricity  as  the  charged  body. 

But  the  metal  and  the  rubber  differ  in  one  very  important 
respect.  If  the  metal  ball  is  touched  at  any  one  point  by  a  wire 
or  by  the  hand,  and  is  thus  connected  to  the  earth  through  the 
wire  or  the  human  body,  the  whole  ball  loses  its  charge.  But 
an  electrified  rubber  or  glass  rod  is  not  completely  discharged 
unless  every  point  of  the  rod  is  touched  with  the  wire  or  with  the 
hand.  That  is,  electricity  moves  freely  from  point  to  point  of 
the  metal,  but  does  not  move  readily  along  rubber  or  glass.  This 
difference  is  further  shown  by  joining  the  electrified  mounted 
ball  with  a  second  mounted  ball,  first,  by  a  glass  or  rubber  rod, 
and  second,  by  a  metal  rod  supported  by  a  rubber  handle. 
The  electrification  is  transmitted  or  conducted  along  the  metallic 
connections  but  not  along  the  glass  or  the  rubber  connection. 
Metals  are  thus  seen  to  be  good  conductors  of  electrification  or 
electricity,  while  glass  and  rubber  are  poor  conductors  or  insula- 
tors. It  is  now  apparent  that  the  purpose  of  mounting  a  metal 
body  on  a  glass  or  rubber  support  is  to  insulate  it  from  the  earth 
and  other  bodies. 

Experiments  show  that  no  substance  is  a  perfect  insulator,  and 
likewise  that  no  substance  is  a  perfect  conductor  of  electricity. 
The  best  insulators  are  amber,  rubber,  sulphur,  shellac,  glass, 
porcelain,  quartz,  air,  silk,  etc.;  the  best  conductors  are  the 
metals,  acids,  moist  earth,  etc.  Dry  wood,  paper,  cotton  and 
linen  thread,  etc.,  are  semi-conductors. 

Gilbert  called  such  substances  as  amber,  rubber,  glass,  sealing  wax, 
quartz,  etc.,  electrics,  after  the  Greek  word  for  amber  (tfKcKrpov).  It  was 
known  to  the  ancient  Greeks  that  amber,  when  rubbed,  acquired  the  strik- 
ing property  of  attracting  pith,  straw,  and  other  light  bodies,  but  up  to 
1600  this  was  an  isolated  fact  regarded  as  peculiar  to  amber  and  jet. 


ELECTROSTATICS  337 

Gilbert  showed  that  many  other  substances  acted  like  amber  when  rubbed, 
and  hence  he  called  such  substances,  electrics,  or  amber-like  bodies.  Gilbert 
failed  to  find  the  same  property  in  metals,  when  he  rubbed  them,  and  hence 
he  called  them  non-electrics.  It  was  not  until  many  years  later  (1736)  that 
Stephen  Gray,  another  Englishman,  showed  that  some  substances  were 
good  conductors  of  electricity  and  other  substances  bad  conductors  or  in- 
sulators. It  was  then  possible  to  show  that  a  metal  body  is  readily  electri- 
fied by  friction,  provided  the  metal  is  supported  on  an  insulating  stand. 
After  this  discovery  the  terms  "electrics"  and  "non-electrics"  lost  their 
meaning,  and  in  the  present  literature  they  have  only  a  historical  interest. 

393.  Electrification  by  Electrostatic  Induction. — If  an  insulated 
conductor,  A,  be  electrified,  say  positively,  and  brought  near  B, 
a  second  insulated  conductor, 
B  becomes  electrified.     This  is 
shown  by  the  repulsion  of  the 
small    pith    ball   electroscopes 
which  are  attached  to  each  end 
of  B.     By  bringing  a  suspended 
gilded  pith  ball  in  contact  with 
A  and  thus  charging  the  ball  FIQ.  253. 

with    positive    electricity,    we 

can  test  the  charges  on  B.  It  is  found  that  the  near  end 
of  B  attracts,  and  the  far  end  repels  the  pith  ball;  that  is,  the 
electrification  on  B  is  of  two  kinds,  the  far  end  having  the 
same  kind  as  that  on  A,  and  the  near  end  having  the  opposite 
kind  to  that  on  B.  If  A  is  now  moved  to  a  distance  from 
B,  B  is  no  longer  charged,  but  becomes  charged  again  when 
A  is  brought  back.  If  B  is  now  joined  to  the  earth  by  a 
wire  or  by  the  hand,  the  charge  at  the  far  end  of  B  disap- 
pears, but  the  charge  on  the  near  end  remains.  The  charge  on 
the  near  end  is  called  a  "  bound  charge,"  while  that  on  the  far  end 
is  called  a  "free  charge."  The  "free  charge"  is  one  that  escapes 
when  joined  by  a  conductor  to  the  earth,  while  the  "bound 
charge"  does  not  so  escape,  because  it  is  attracted  by  a  charge 
of  the  opposite  kind.  If  the  connection  of  B  with  the  earth  is 
broken,  and  A  removed,  it  is  found  that  the  charge  on  B  dis- 
tributes itself  over  the  conductor  and  is  "free."  The  conductor 
B  is  then  electrified  oppositely  to  A,  while  the  charge  on  A  is  not 
diminished.  The  above  process  is  called  charging  a  body  by 
electrostatic  induction  or  influence. 
22 


338  ELECTRICITY  AND  MAGNETISM 

394.  Theories  of  Electricity. — It  has  been  stated  that  electri- 
fication is  assumed  to  be  due  to  an  agent  called  "electricity." 
Various  theories  have  been  held  as  to  the  nature  of  electricity. 

One  of  the  first  theories  was  that  due  to  Benjamin  Franklin 
and,  although  stated  a  hundred  and  sixty  years  ago,  it  is  still, 
in  the  essentials,  one  of  the  most  consistent  theories  of  electricity. 
Franklin  assumed  that  there  is  an  "electrical  matter/'  probably 
consisting  of  very  fine  particles,  so  light  as  to  be  practically 
imponderable  or  without  weight,  and  that  this  electrical  matter 
flows  most  freely,  that  is,  it  is  a  "fluid."  This  electrical  fluid 
is  distributed  throughout  all  bodies,  and  each  body  has  naturally 
a  certain  normal  amount  of  it.  If  more  than  this  normal  amount 
is  added  to  a  body,  the  body  is  positively  electrified;  if  the  body 
by  any  means  has  less  than  its  normal  amount  of  the  fluid,  the 
body  is  negatively  electrified.  Further,  "electrical  matter 
differs  from  common  matter  in  that  the  parts  of  electrical  matter 
naturally  repel  each  other,"  but  they  attract  ordinary  matter. 
Thus  the  process  of  electrifying  rubber  by  friction  is  one  in  which 
the  fur  gets  more  than  its  normal  amount  of  the  electrical  fluid, 
and  the  rubber  less  than  its  normal  amount,  while,  in  rubbing 
glass  with  silk,  the  glass  gains  electrical  fluid  at  the  expense  of 
the  silk.  To  electrify  a  body  positively  is  thus  simply  to  trans- 
fer from  a  second  body  a  portion  of  its  electrical  fluid,  and 
the  second  body  will  then  have  a  deficit  or  will  be  negatively 
electrified. 

Another  fluid  theory  of  electricity,  that  has  been  widely  held, 
is  Symner's  two-fluid  theory  of  electricity.  Symner  assumed 
that  there  are  two  electrical  fluids,  a  positive  fluid  and  a  negative 
fluid.  In  its  neutral  or  unelectrified  condition,  a  body  has  equal 
quantities  of  these  two  fluids;  when  a  body  is  electrified  posi- 
tively, it  has  more  positive  than  negative  fluid;  and  when  electri- 
fied negatively,  it  has  more  negative  than  positive  fluid.  It  is 
further  assumed  that  the  two  fluids  attract  each  other.  It  is 
evident  that  the  above  fluid  theories  are  equivalent  to  each  other, 
if  we  simply  suppose  Symner's  negative  fluid  is  the  "deficit  of 
the  positive  fluid."  The  Franklin  theory  has  the  advantage  of 
assuming  only  a  single  fluid,  and  is  more  nearly  in  accord  with 
the  present  electron  theory  of  electricity. 

The  electron  theory  of  electricity  is  Franklin's  one-fluid  theory 


ELECTROSTATICS 


339 


extended  and  made  much  more  precise  so  as  to  account  for 
numerous  phenomena  recently  discovered.  According  to  this 
theory  electrification  is  due  to  negatively  charged  particles, 
called  electrons  or  corpuscles,  which  are  all  precisely  similar  but 
very  much  smaller  than  the  smallest  atoms.  In  its  natural  un- 
electrified  condition  a  body  has  a  certain  number  of  electrons; 
when  it  has  more  than  this  normal  number,  the  body  is  nega- 
tively electrified,  and,  when  it  has  less  than  the  normal  number, 
it  is  positively  electrified.  Different  lines  of  research  have  shown 
that  the  mass  of  an  electron  must  be  about  1/1800  of  the  mass  of 
a  hydrogen  atom.  It  seems  probable  that  in  a  non-conductor 
most  of  the  electrons  are  associated  with  or  bound  to  atoms  and 
possibly  vibrate  or  rotate  about  the  centers  of  atoms,  as  planets 
rotate  about  the  sun;  but  in  conductors  most  of  the  electrons  are 
dissociated  from  atoms  and  are  capable  of  moving  about  freely, 
thus  accounting  for  the  flow  of  electricity  in  conductors.  While 
the  body  of  evidence  for  the  electron  theory  in  some  form  is  very 
great,  the  mechanism  of  the  attraction  between  electrons  and 
atoms  which  have  less  than  the  normal  number  of  electrons 
remains  as  yet  unexplained,  and,  to  allow  for  this  difficulty,  it 
is  still  customary  to  speak  of  an  atom  as  having  a  charge  or 
"nucleus"  of  positive  electricity  which  it  cannot  lose. 

The  fluid  theory  of  electricity  has  in  some  form  been  used  so 
long  as  a  working  hypothesis,  that  the  terms  of  electrical  science 
are  based  on  the  concept  of  a  fluid.  But  in  using  such  words 
as  "flow,"  "current,"  etc.,  we  do  not  commit  ourselves  to  any 
particular  theory. 

395.  Gold-leaf  Electroscope. — The  most  sensitive 
and  generally  useful  means  of  detecting  electrifica- 
tion is  the  gold-leaf  electroscope.  In  its  usual 
form,  it  consists  of  two  pieces  of  gold  leaf  hung 
beside  each  other  from  the  lower  end  of  an  in- 
sulated metal  rod.  The  upper  end  of  the  rod  termi- 
nates in  a  ball  or  a  plate.  The  gold  leaves  are 
enclosed  in  a  case  made  wholly  or  partly  of  glass,  for 
protection  from  air  currents  and  so  that  the  move-  FIO.  25*- 
ment  of  the  leaves  can  be  observed.  When  the  case 
is  largely  of  glass,  strips  of  tin  foil  are  often  pasted  on  the  glass 
and  connected  through  the  base  to  earth  for  "  screening  "fc( §3 96). 


340  ELECTRICITY  AND  MAGNETISM 

If  the  plate  of  the  electroscope  is  electrified  by  contact  with 
a  charged  body,  the  leaves,  being  charged  with  like  kinds  of 
electricity,  diverge,  and  stay  apart  until  the  electroscope  is  dis- 
charged by  connection  with  the  earth. 

The  more  usual  method  of  charging  the  electroscope  is  by 
electrostatic  induction.  When  a  body  which  is  charged  posi- 
tively is  brought  near  the  plate,  che  latter  becomes  charged  with 
a  "bound"  negative  charge  and  the  leaves  with  a  "free"  posi- 
tive charge.  The  free  charge  escapes  when  connection  is  made 
to  earth,  and  the  leaves  collapse.  The  earth  connection  is  now 
broken  and  the  electrified  body  is  then  removed,  thus  freeing 
the  "bound"  negative  charge.  This  spreads  over  the  electro- 
scope and  the  leaves  diverge.  The  electroscope  is  thus  charged 
negatively,  that  is,  oppositely  to  the  inducing  charge.  If  now  a 
positive  charge  is  again  brought  up,  the  leaves  collapse  but,  if  a 
negative  charge  is  brought  up,  the  leaves  diverge  still  further. 
Hence,  if  we  know  the  kind  of  charge  on  an  electroscope,  we  can 
determine  the  kind  of  charge  on  a  body.  If  the  leaves  first  con- 
verge as  the  body  is  brought  up,  then  the  body  is  charged  oppo- 
site to  the  electroscope;  if  the  leaves  diverge  as  the  body  is 
brought  up,  then  the  body  is  charged  with  the  same  kind  of 
electricity  as  the  electroscope. 

In  a  modified  form  of  the  gold-leaf  electroscope  (Fig.  255),  a  single  strip  of 
gold  leaf  hangs  along  a  brass  plate.  The  exact  divergence  of  the  gold  leaf 
from  the  plate  is  easier  to  determine  than  the 
amount  of  divergence  of  two  leaves,  and  so  this 
form  is  better-  adapted  for  making  measurements. 
The  figure  also  shows  devices  to  secure  the  highest 
insulation.  The  brass  plate  P  with  its  gold-leaf 
strip  L  is  supported  separately  by  a  sulphur  bead 
S,  and  connection  for  charging  is  made  by  a  special 
charging  wire.  The  latter  is  a  wire  bent  with  two 
right  angles,  and  fixed  so  that,  by  turning  it,  con- 
nection between  the  gold  leaf  and  the  upper  disk 
can  be  made  or  broken. 

396.  Electricity  Confined  to  Surface  of  Con- 
ductors.— A  very  important  law  in  the  distri- 
bution of  electrification  on  conductors  is  that  it  is  all  on  the  sur- 
face of  the  conductor.     One  method  of  showing  this  is  by  means 
of  a  hollow  conductor  in  which  there  is  a  small  opening.     The 


. 


ELECTROSTATICS 


341 


conductor  is  insulated  and  charged.  If  a  small  metal  plate  with 
an  insulating  handle,  called  a  "proof  plane,"  be 
now  brought  in  contact  with  the  various  parts  of 
the  surface  of  the  conductor,  and  then  tested  by 
bringing  it  to  the  gold-leaf  electroscope,  it  is  found 
to  be  charged.  But  if  it  be  touched  on  the  inside 
and  brought  to  the  electroscope  there  is  no  charge. 
That  is,  there  is  no  charge  on  the  inside  of  a  con- 
ductor, unless  the  charge  is  insulated  from  the 
conductor. 

Another  experiment  showing  this,  is  to  charge  an 
insulated  metal  body,  and,  after  carefully  introduc- 
ing it  through  the  opening  of  the  hollow  conductor, 
to  touch  it  to  the  inside  of  the  hollow  conductor;  on 
removing  and  testing  the  body,  it  is  found  to  be 
completely  discharged,  its  charge  being  now  found 
on  the  outside  of  the  conductor. 

Another  experiment  is  shown  in  Fig.  257.    A,  a  metal  sphere  on  an 
insulating  stand,  is  charged.     Two  insulated  hemispherical  metal  cups, 
B  and  C,  are  arranged  so  as  to  completely 
enclose  and  touch  A.     When  B  and  C  are  re- 
moved, it  is  found  that  A  is  free  from  any 
charge  and  all  the  charge  is  on  B  and  C. 

Still  another  experiment  showing  the  same 
law,  is  to  put  a  sensitive  electroscope  inside 
a  finely  woven  wire  cage,  connecting  it  with 
the  cage.  The  insulated  cage  can  now  be 
strongly  electrified  and  still  the  electroscope 
will  show  no  charge  on  the  inside.  Faraday 
constructed  a  large  metallic  covered  box  which 


FIG.  256. 


=€)O 

T 


Fio.  257.  Fia.  258. 

ha  insulated,  and  into  it  he  carried  his  most  sensitive  electroscopes.  He 
found  that  these  showed  no  effects,  even  when  spark  discharges  took  place 
from  the  outside.  Experiments  with  the  thinnest  of  films  show  that  the 
electrification  is'^always  on  the  surface. 


342  ELECTRICITY  AND  MAGNETISM 

The  above  experiments  also  show  that  a  body  can  be  shielded 
from  outside  electrical  disturbances  by  surrounding  it  with  a 
metal  case.  This  is  done  frequently  with  measuring  instru- 
ments, especially  with  'electroscopes  and  electrometers.  The 

explanation  of  the  above  facts  in  terms 
of  lines  of  force  will  be  given  later  (§399) . 
397.  Distribution  of  Electrification  on 
Conductors.  Effect  of  Points. — We  can 
investigate  the  distribution  of  electrifi- 
cation on  the  different  parts  of  a  con- 
ductor by  means  of  a  "proof  plane"  and 
a  gold-leaf  electroscope.  For  this  pur- 
pose take  an  egg-shaped  conductor  and 
charge  it.  Touch  the  proof  plane  to 
various  parts  of  the  conductor  and  then 
test  the  proof  plane  by  the  electroscope 
(Fig.  259).  It  is  found  that  the  deflec- 
tion of  the  electroscope  is  greatest  when  the  "proof  plane"  has 
been  in  contact  with  the  pointed  end  of  the  conductor,  and 
least  when  it  has  been  in  contact  with  the  flat  parts  of  the  con- 
ductor; or,  in  general,  that  the  electrification  is  greatest  at  parts  of 
greatest  curvature. 

The  curvature  of  a  sharp  point  approaches 
infinity,  and  it  follows  from  the  above,  that  -*- 
the  electrification  on  such  a  point  should 
become  very  great.  That  this  is  so  is  shown 
by  the  fact  that  an  insulated  conductor  sup- 
plied with  needle  points  discharges  itself  almost 
immediately.  Also,  if  a  needle  point  be  held 
toward  a  charged  conductor,  the  conductor  loses 
its  charge  almost  immediately.  The  induced 
electrification  on  the  point  is  so  great  that  it  somehow  breaks 
y—  — ^  across  and  discharges  the  conductor.  An 

interesting  device  for  showing  the  discharge 
from  points  is  "  the  electric  wheel ' '  (Fig.  260) . 
It  consists  of  a  series  of  pointed  wires  in- 
serted   horizontally  into  a  metal  ball,  which  is    balanced   on 
a    steel  point,  the  pointed   wires  being  arranged  as  shown,  in 


ELECTROSTATICS 


343 


a  "whirl."      The  discharge  from  the  points  causes  a  reaction 

which  drives  the  wheel  around  rapidly. 

A  metal  rod  carrying  a  row  of  metallic 

points  (Fig.  261),  and  called  a  "comb," 

is  used  in  static  electrical  machines  to 

collect  the  charges  from  the  revolving 

glass  plates  across  a  short  air  space       ^''         R 

(see  §409). 

398.  Fields  and  Lines  of  Electric 
Force. — The  experiments  which    we  _^ 

have  described  above  can  all  be  ex-  / 
plained,  if  we  assume  that  one  electric 
charge  acts  on  another  "at  a  dis- 
tance," that  is,  directly  across  space 
without  the  action  of  an  intervening 
medium.  Thus,  to  explain  electro- 
static induction,  we  might  say  that 
the  positive  charge  on  a  body  A  repels  FIQ.  202. 

the   positive   charge  in  the  gold-leaf 

electroscope  to  the  leaves,  and  attracts  the  negative  electricity 
to  the  plate,  in  this  account  making  no  mention  of  any  inter- 
vening medium.  But  a  simple  experiment  shows  that  the  inter- 
vening medium  cannot  be  neglected.  If  we  introduce  a  thick 
plate  of  hard  rubber,  R,  between  the  inducing  charge  on  A,  (Fig. 
262),  and  the  electroscope  E,  we  see  that  the  leaves  come  nearer 

together,  and  when  the  hard  rubber 
plate  is  removed,  they  go  back  to 
their  original  position.  The  action  is 
similar  with  plates  of  sulphur,  shellac, 
glass  and  other  insulators.  That  is, 
we  see  that  the  inductive  action  of  A 
upon  the  electroscope,  depends  upon 
the  intervening  medium.  Similar 
phenomena  led  Michael  Faraday  to 
a  study  of  the  insulating  medium. 
(See  §413.) 

Faraday  could  not  conceive  that  one  body  could  act  upon 
another  otherwise  than  by  a  push  or  a  pull  through  an  inter- 
vening body  or  medium  of  some  kind.  Like  Newton  in  the  case 


344 


ELECTRICITY  AND  MAGNETISM 


of  gravitation  he  could  not  think  of  "  a  body  acting  where  it  was 
not."  Faraday,  therefore,  called  an  insulating  medium  a  dielectric, 
a  word  suggesting  electric  action  through  or  across  the  medium, 
because  he  thought  of  the  electric  action  as  due  to  stresses  in  the 
intervening  insulators.  In  accordance  with  this  idea,  he  called 
the  space  about  an  electrified  body  a  field  of  electric  force,  or  an 
electric  field.  We  can  think  of  an  electric  field  as  a  region  in 


Fio.  264a 

which  there  are  electric  stresses,  and  these  stresses  can  be  indi- 
cated by  lines  or  tubes.  A  line  of  electric  force  is  the  path  along 
which  a  small  positive  charge  tends  to  move.  Thus,  about  an 
electrified  sphere  there  are  radial  lines  or  tubes,  indicating  the 
direction  of  the  stress  in  the  field.  We  can  map  out  these  lines 
of  stress  by  taking  a  small  positively  electrified  body,  and  noting 
the  direction  of  the  force  on  the  body  at  each  point.  In  the  case 

of  a  charged  sphere  hung  in  air 
at  a  very  great  distance  from  all 
other  conductors,  these  lines  ap- 
parently disappear  where  the 
forces  become  too  small  to  detect, 
but  if  there  are  two  conductors, 
(Fig.  264a),  one  charged  posi- 
tively and  the  other  negatively, 
many  of  the  lines  that  start 
from  the  positive  charge  will  be 


Fia.  2646. 


ELECTROSTATICS 


345 


found  to  terminate  in  the  negative  charge.    Figure  2646  shows 
the  lines  of  force  of  the  field  due  to  two  equal  positive  charges. 

The  mechanism  of  an  electric .  field  is  not  yet  wen  understood 
Faraday  regarded  a  line  or  tube  of  electric  force  as  a  chain  of  "polarized" 
particles  of  the  intervenir^  medium.  By  a  "polarized"  body  is  meant  a 
body  which  has  equal  buj  opposite  properties  at  its  two  ends  or  sides. 
Thus  a  bar  magnet  with  its  equal  north  and  south  poles,  or  a  metal  sphere 
with  equal  positive  and  negative  induced  charges  is  polarized. 

According  to  the  electron  theory  each  polarized  particle  in  a  dielectric 
consists  of  an  atom  and  its  associated  electrons.  Being  oppositely  charged, 
they  will  tend  to  move  in  opposite  directions  in  an  electric  field,  but  the 
separation  will  be  very  slight  and  will  be  limited  by  the  attractions  between 
them.  The  separation  will  be  greater  the  greater  the  intensity  of  the  field. 
We  can  thus  form  an  instructive  mental  picture  of  what  takes  place  in  a 
dielectric,  but  this  will  not  apply  to  an  electric  field  in  a  vacuum  and  for 
this  we  have  at  present  no  explanation. 

399.  Faraday's  Ice-pail  Experiment. — Lines  of  electric  force 
start  from  positive  charges  of  electricity  and  terminate  in  nega- 
tive charges  and  there  is  a  negative  charge  somewhere  corre- 
sponding to  every  positive  charge.  An  experiment  due  to  Fara- 
day (Fig.  265)  proves  this  to  be  the  case.  It  is  known  as  "the 
ice-pail  experiment/'  because,  when  it  was  first  performed,  a 
metal  ice-pail  was  used  as  the  most  convenient  vessel  at  hand. 

A  metal  vessel  B  with  a  narrowed  opening  is  insulated  and 
connected  by  a  wire  with  the  uncharged  gold-leaf  electroscope 
E.  The  ball  A,  hung  by  a  silk  thread  and  charged  positively,  is 
let  down  into  B,  but  does  not  touch  B;  the  electroscope  becomes 
charged  positively,  and  on  the  insde  of  B  we  have  an  induced 
negative  charge.  If  A  is  removedi 
without  touching  B,  the  electro- 
scope leaves  contract,  showing  that 
the  two  induced  charges  unite  and 
exactly  neutralize  each  other.  Now 
if  A  is  again  put  inside,  the  leaves 
of  the  electroscope  again  diverge; 
and  if  now  the  ball  is  touched  to 
B,  there  is  no  change  in  the  di- 
vergence of  the  electroscope.  When  FIQ.  265. 
A  is  taken  out,  it  is  found  to  be 

discharged.      Evidently    the    positive    charge    on    A    exactly 
neutralized  the  induced  negative  charge  on  the  inside  of  B,  and 


346 


ELECTRICITY  AND  MAGNETISM 


Fio.  266. 


thus  B  and  E  remain  charged  positively.  Hence  the  induced 
charge  is  equal  and  opposite  to  the  inducing  charge. 

Since  for  every  charge  there  is  an  equal  opposite  charge  some- 
where on  the  surrounding  conductors  (walls  of  room,  earth,  etc.), 
we  must  think  of  lines  of  electric  force  as 
always  connecting  these  opposite  charges, 
and  also  as  being  in  a  state  of  tension,  so 
that  they  tend  to  contract  and  draw  the 
two  opposite  charges  together.  The  lines 
of  force  also  seem  to  repel  each  other,  as 
appears  from  the  figures.  This  shows  a 
lateral  pressure  in  the  medium  which  is  very 
important  in  the  theory  of  dielectric  action. 
Electric  repulsion  is  by  this  means  re- 
solved into  two  attractions  in  opposite 
directions.  The  repulsion  between  the  two  gold  leaves  in 
the  electroscope  is  due  to  the  tension  of  the  lines  of  force 
between  the  charges  on  the  leaves  and  the  induced  charges 
on  the  walls  of  the  case  (Fig.  266).  The  sensitiveness  of 
the  gold-leaf  electroscope  is  thus  changed  by  the  presence  of 
these  neighboring  conducting  walls.  The  attraction  of  an  elec- 
trified body  for  an  uncharged  conductor  can  now  be  seen  to  be 
due  to  the  tension  of  lines  of  electric  force.  When  the  neutral 
body  B  is  brought  into  the  electric  field  of  a  positively  charged 
body  A,  there  is  induced  in  B  a  negative  charge  on  the  near  side 
and  a  positive  charge  on  the  far  side,  that  is,  lines  of  force  con- 
nect A  and  B,  as  in- 
dicated in  Fig.  267, 
and  it  is  the  result- 
ant of  the  pulls  of  all 
these  lines  that  causes 
the  attraction. 

We  have  already 
seen  that,  in  electri- 
fication by  friction, 
the  fur  is  electrified 

positively  at  the  same  time  that  the  rubber  is  electrified  nega- 
tively. By  the  "ice-pail"  apparatus  it  can  be  shown  that 
equal  quantities  of  the  two  kinds  are  produced  in  the  case 


FIG.  267. 


ELECTROSTATICS 


347 


Fio.  268. 


of  friction.  Fasten  a  small  piece  of  fur,  F,  on  an  insulating 
handle,  and  rub  the  fur  with  a  rubber  rod,  R,  (Fig.  268).  If 
both  are  inside  the  "ice-pail,"  the  gold  leaves  indicate  no 
charge;  but  when  either  is  taken 
out  there  is  a  deflection.  Hence  the 
electrification  on  the  fur  is  equal 
and  opposite  to  that  on  the  rubber. 
The  experiments  with  the  ice- 
pail  apparatus  show  that  in  the 
charging  and  discharging  of  bodies 
there  is  no  creation  or  destruction 
of  electricity,  but  simply  transfers 
of  electricity.  The  total  quantity 
of  electricity  remains  unchanged. 
This  fact  is  known  as  the  conserva- 
tion of  electricity,  and  is  in  accord 
with  the  fluid  theories,  and  also  with  the  modern  electron  theory. 

400.  Energy  of  Charged  Bodies. — The  energy  which  an  elec- 
tric charge  represents  comes  from  the  work  done  in  separating 
the  two  kinds  of  electricities,  and  is  equivalent  to  it.     Thus, 

when  we  rub  a  rod  of  rubber  with  fur,  the  elec- 
trification is  in  no  way  proportional  to  the  fric- 
tion. A  very  light  but  complete  contact  is  in 
fact  all  that  is  needed  and  a  perfect  contact  is 
the  only  aim  in  rubbing  the  bodies  together. 
But,  to  separate  the  fur  with  its  positive  charge 
from  the  negatively  charged  rod,  work  must  be 
done  against  the  mutual  attraction  of  the  two 
charges,  or,  using  Faraday's  concept  of  lines- of 
force,  the  work  is  done  in  setting  up  stresses  in 
the  intervening  medium,  these  stresses  being 
represented  by  the  lines  or  tubes  of  force 
stretching  between  the  fur  and  the  rod  (Fig. 
269).  When  the  two  charges  come  together 
again,  the  lines  or  tubes  contract,  and  do  work.  The  energy  on 
this  view  is  analogous  to  that  of  stretched  elastic  bands  con- 
necting two  bodies. 

401.  Law  of  Electrical  Force. — The  law  which  states  how  the 
force  between  two  electrical  charges  depends  upon  the  charges 


Fio.  269. 


348 


ELECTRICITY  AND  MAGNETISM 


and  upon  the  distances  between  them,  was  first  published  by  Cou- 
lomb in  1785  and  hence  is  known  as  Coulomb's  law  of  electric 
force.  The  law  states  that  the  force  between  two  electrical  charges 
varies  (a)  inversely  as  the  squares  of  their  distance  apart,  and  (b) 
directly  as  the  product  of  the  two  electrical  charges.  This  is  ex- 
pressed by  the  formula, 

T?  -  *  "' 

K^r* 

where  F  is  the  force,  q  and  g'  the  two  charges,  r  'their  distance 
apart,  and  l/K  a  constant  depending  upon  the  units  used  and 
also  upon  the  intervening  medium.  If  we  use  the  dyne  as  the 
unit  of  force,  the  centimeter  as  the  unit  of  length,  and  define 
the  unit  charge  or  unit  of  electric  quantity  q,  as  follows:  Unit 
electric  quantity  is  that  quantity  which  at  one  centimeter  distance  in 
air  exerts  a  force  of  one  dyne  on  an  equal  quantity;  then  the  formula 
for  charges  in  air  becomes: 


In  the  case  of  air,  K  has,  by  the  definition  of  unit  electric  quantity, 
the  value  unity.  For  other  intervening  media  or  dielectrics, 
^  the  more  general  formula  must  be  used. 

The  values  of  K  for  several  dielectrics  are 

as  follows:  (compare  §413) 

Air 1.00 

Petroleum  oil 2.07 

Turpentine 2 . 23 

Distilled  water 75.  + 

Coulomb  arrived  at  the  above  law  by  ex- 
periments with  his  torsion  balance  (Fig.  270) 
similar   to  those  by  which  he  discovered 
the  law  of  magnetic  forces  (§372),  but  the 
best  proof  of  the  law  is  the  indirect  one, 
that  it  is  the  only  relation  that  explains 
exactly  electrostatic  phenomena.     In  fact, 
Henry  Cavendish  before  1785,  thus  estab- 
lished the  law  for  his  own  use,  though  his 
papers  were  first  published  nearly  a  century  later,   long  after 
others  had  reached  the  same  results.     This  proof  is  based  on 
the  experimental  fact  that  at  any  point,  0,  inside  a  hollow  con- 


ELECTROSTATICS  349 

ducting  sphere  there  is  no  electric  force  from  charges  on  the 
surface  of  the  sphere.  Draw  straight  lines  through  0,  dividing 
the  whole  sphere  into  pairs  of  cones  of  small  angular  opening, 
and  with  a  common  apex  at  0,  and  with  bases  Si  and  S2,  etc., 
cut  out  of  the  spherical  surface,  the  heights  of  these  cones 
being  rlt  r2,  etc.  Let  Qv  and  Q2  be  the  charges  on  the  bases 
St  and  S2.  Since  the  electricity  is  distributed  uniformly  on  the 
sphere,  the  charges  Qt  and  Q3  are  proportional  to  the  areas 

O         Sf 
Sj^  and  $3,  or  ^-  =  -^.     But  the  cones  have  the  same  angle  and 

V2  02 

hence  the  bases  o\  and  S2  are  to  each  other  as  the  squares  of  the 

heights,  or  ~  =  -^,  or  ^~  =  St     Thus,  the  forces  being  equal 
02      r**        rv      TV 

and  opposite,  the  resulting  force  at  0  on  a  test  unit  must  be  zero, 

if  the  inverse  square  law  holds.     Since  these  equations  hold  for 

all  values  of  rt  and  r2,  it  is  evident  that  the 

inverse  square  law  is  the  only  one  meeting 

the  conditions  of  this  experiment.     Maxwell 

and  others  have  repeated  this  experiment 

with  very  sensitive  electroscopes  so  that  the 

law  of  the  force  varying  inversely  as  the 

square  of  the  distance  is  one  of  the  best 

established  laws  of  electrostatics. 

402.  Electrical    Potential. — To    describe  Flo  271 

and  explain  the  movement  of  electricity 
the  terms  "electrical  potential"  and  "differ  ence  of  electrical 
potential"  are  used.  Thus  if  we  join  two  conductors  A  and 
B  by  a  wire  and  find  that  there  is  a  flow  of  electricity  from 
A  to  B,  we  ascribe  this  to  a  "  difference  of  electrical  potential" 
between  the  two  bodies,  and  say  that  there  is  a  flow  from 
A  to  B  because  A  is  at  "the  higher  potential"  and  B  at 
"the  lower  potential."  In  the  case  of  electrostatic  charges,  such 
as  we  have  been  describing,  this  flow  or  current  is  only  momen- 
tary, because  the  two  bodies  come  in  an  instant  to  the  same 
"potential."  By  means  of  batteries  and  dynamos,  as  we  shall 
see  later,  it  is  possible  to  maintain  a  continuous  "  difference  of 
potential"  and  hence  a  continuous  electric  flow  or  current  be- 
tween two  points  of  a  conductor.  The  movement  of  a  charged 
body  from  one  point  to  another  point  in  an  electric  field  is  also 
described  as  due  to  a  "  difference  of  electrical  potential  between 


350  ELECTRICITY  AND  MAGNETISM 

the  two  points"  of  the  field.  The  positively  charged  body 
tends  to  move  from  A  to  B,  because  the  point  A  is  at  "  a  higher 
electrical  potential"  than  the  point  B. 

Potential  as  used  above  is  very  analogous  to  pressure  in  fluids. 
Thus  flow  of  a  gas  takes  place  from  a  tank  of  higher  pressure  to 
a  tank  of  lower  pressure  when  the  tanks  are  connected,  and  the 
flow  continues  until  the  pressures  are  equalized.  Another  very 
useful  analogy  is  that  of  level  in  liquids.  A  liquid  tends  to  flow 
from  points  of  higher  level  to  points  of  lower  level;  to  maintain 
the  flow,  the  difference  in  level  must  be  maintained.  When  a 
liquid  flows  from  a  higher  level  to  a  lower  level,  it  loses  some  of  its 
potential  energy  by  transformation  into  energy  of  some  other 
form.  In  fact  the  potential  energy  of  a  system  always  tends  to 
a  minimum.  (§107.) 

Let  us  apply  this  to  the  case  of  a  charge  in  an  electric  field. 
Consider  the  electric  field  (Fig.  263,  §398),  about  a  positively 
charged  sphere  in  air  all  other  bodies  being  supposed  to  be  at 
indefinitely  great  distances.  A  positive  unit  charge  at  a  point 
x  in  this  electric  field  has  a  certain  potential  energy,  Vx,  this 
being  the  number  of  ergs  of  work  that  the  charge  can  do  by 
moving  under  the  action  of  the  forces  of  the  field  from  x  to 
infinity  (that  is,  completely  out  of  the  field).  It  is  also  equal  to 
the  work  that  is  required  to  move  the  unit  charge  from  an 
infinite  distance  up  to  the  point  x.  The  potential  energy  of  a 
positive  unit  charge  at  a  point  x  we  call  "the  electrical  potential 
at  the  point  z,"  and  represent  it  by  Vx.  In  the  same  way, 
the  electrical  potential  at  a  point  y,  or  Vv,  is  the  potential 
energy  of  unit  charge  at  y.  The  potential  energy  of  a  charge 
in  an  electric  field,  as  in  other  cases  of  potential  energy  (§63), 
evidently  depends  only  on  the  final  position  of  the  charge,  and 
not  upon  the  path.-  That  is,  each  point  in  an  electric  field  has 
a  definite  electrical  potential.  Hence  the  difference  of  electrical 
potential  of  two  points  x  and  t/^  that  is,  Vy—  Vx,  is  a  definite 
quantity,  and  is  defined  as  follows:  The  difference  in  electrical 
potential  between  two  points  x  and  y  in  an  electric  field  is  equal 
to  the  number  of  ergs  required  to  move  unit  positive  charge  from 
the  point  x  to  the  point  y.  If  this  work  is  positive,  that  is,  if 
external  work  must  be  done  on  the  positive  charge  to  move  it, 
then  the  potential  of  y  is  higher  than  the  potential  of  x. 


ELECTROSTATICS  351 

We  thus  see  that  it  is  in  accordance  with  the  principle  of  mini- 
mum potential  energy  that  a  positive  charge  tends  to  move 
from  a  point  of  higher  electrical  potential  to  a  point  of  lower 
electrical  potential.  In  the  above  we  have  assumed  that  the 
intensity  of  the  electric  field  was  not  appreciably  changed  by 
the  presence  of  the  unit  charge. 

403.  Zero  Potential,  Positive  and  Negative  Potential. — In  the 
case  of  water  levels  we  choose  some  arbitrary  level  as  a  reference 
or  zero  level,  the  level  of  the  sea  being  so  chosen  by  universal 
agreement.  All  levels  above  sea-level  are  marked  positive  or 
plus  (+),  and  all  levels  below  sea-level  are  marked  negative  or 
minus  (  — ).  In  an  analogous  way,  the  electrical  potential  of 
the  earth  is  taken  as  the  zero  potential.  Since  the  earth  is  a 
conductor,  all  points  on  it  for  electrical  equilibrium  are  at  the 
same  potential;  otherwise  there  would  be  an  electrical  flow  until 
equilibrium  was  reached.  A  body  A  is  thus  at  a  positive  elec- 
trical potential  when  positive  electricity  tends  to  flow  from  A 
to  the  earth;  and  in  the  same  way,  a  body  B  is  at  a  negative 
electrical  potential  when  positive  electricity  tends  to  flow  from 
the  earth  to  the  body  B.  To  say  a  conductor  has  a  "free"  posi- 
tive (or  negative)  charge  is  equivalent  to  saying  that  it  is  at 
a  positive  (or  negative)  potential. 

The  electrical  potential,  V,  of  a  point  or  of  a  body  with  refer- 
ence to  the  earth  is  therefore  equal  to  the  number  of  ergs  required 
to  move  unit  positive  electricity  from  the  earth  to  the  point  or 
to  the  body.  If  this  work  is  positive,  that  is,  if  positive  work  is 
done  on  the  test  unit,  then  the  potential  is  positive;  but  if  the 
work  is  done  by  the  test  unit,  then  the  potential  is  negative. 

'404.  Equipotential  Surfaces. — In  an  electrical  field  all  points 
which  have  the  same  potential  lie  on  an  equipotential  surface. 
To  determine  if  two  points  are  on  an  equipotential  surface  is  to 
determine  if  work  is  done  against  electrical  forces  in  the  move- 
ment of  a*  charge  from  the  one  point  to  the  other,  or  whether  a 
test  charge  tends  to  move  from  either  point  to  the  other.  Lines 
of  force  always  cut  an  equipotential  surface  at  right  angles; 
otherwise  there  would  be  a  force  component  along  the  surface, 
which  is  not  possible  if  a  charge  does  not  tend  to  move  along 
the  surface.  Hence  in  the  case  of  the  electric  field  about  a 
single  charge  distant  from  other  charges,  the  equipotential  sur- 


352  ELECTRICITY  AND  MAGNETISM 

faces  are  concentric  spherical  surfaces.  In  the  figures  of  elec- 
tric fields  (§398),  the  equipotential  surfaces  are  indicated  by  the 
dotted  lines  at  right  angles  to  the  lines  of  force.  The  surface 
of  a  conductor  is  evidently  an  equipotential  surface,  if  the 
electric  charges  on  it  are  at  rest;  and  hence  lines  of  force  enter 
and  leave  a  conductor  at  right  angles  to  the  surface. 

406.  Units  of  Quantity  and  of  Potential. — The  unit  of  electric 
quantity  has  been  defined  (§401)  as  the  quantity  which  at  one 
centimeter  distance  in  air  exerts  a  force  of  one  dyne  on  an  equal 
quantity.  This  is  the  c.g.s.  electrostatic  unit  quantity.  For  prac- 
tical measurements  a  much  larger  unit  called  the  "coulomb" 
is  used.  We  can  for  the  present  define  the  coulomb  as  follows: 
1  coulomb  =  3  XlO9  c.g.s.  electrostatic  units  of  quantity. 

The  c.g.s.  electrostatic  unit  difference  of  potential  exists  between 
two  points  when  one  erg  of  work  is  done  in  the  movement  of  a  c.g.s. 
electrostatic  unit  of  charge  from  the  one  point  to  the  other  point. 
Hence  to  move  q  c.g.s.  electrostatic  units  of  electricity  from  a 
point  of  potential  Vl  to  one  of  potential  V2  takes  q(Vt—  FJ  ergs 
of  work,  or 

W  =  q(V,-Vt). 

In  practical  measurements  of  difference  of  potential  the  volt  is 
used  as  the  unit;  1  volt  =  1/300  or  1/3  XlO~3  c.g.s.  electrostatic 
units  difference  of  potential. 

From  the  above,  it  follows  that  the  movement  of  a  coulomb  of 
electricity  against  a  D.  P.  of  one  volt,  represents  3Xl09Xl/3 
X  10~3  ergs  =  107  ergs  =  1  joule  (§55).  Hence 

W (joules)  =Q(coulombs)  X  D.  P.  (volts). 

In  the  special  section  on  electrical  units  (§547)  the  reason  for 
the  choice  of  the  above  practical  units  will  be  discussed. 

406.  Potential  Calculations. — The  difference  of  potential  be- 
tween two  points  a  and  n  due  to  a  charge  Q  is  given  by  tfce  formula 

FO—  Vn  —Ql ),  where  ra  and  rn  are  the  distances  of  a  and 

ya      Tnl 

n  from  Q  and  the  medium  is  for  the  present  assumed  to  be  air. 
For  let  a  and  n  and  Q  be  in  a  straight  line.     (Fig.  272.) 
The  force  on  unit  charge  at  a  is  Q/r20,  and  the  force  at  b,  a  point 
near  a  is  Q/rV     The  average  force  between  a  and  b  can  be  taken 


ELECTROSTATICS  353 

as  Q/rarb.     The  work  done  by  the  field  in  moving  unit  charge 
from  a  to  b  is  then  (rb  —  r0)  Q/rarb.      Hence  it  follows  that 


Va-Vb  =  (r6-ra 
Similarly  for  a  series  of  neighboring  points 


Vb-V6=Q(llrb-l/re) 


Summing  these  up,  we  get 

Va-Vn=Q(l/ra-l/rn). 

If  the  point  n  is  at  an  infinite  distance  then  rn  =  oo  and  l/rn  =0; 
and  hence  the  potential  of  the  point  a  is  Va=Q/ra.  Similarly 
the  potential  at  any  point  x  in  the  field  of  Q  is  Vx  =Q/rx,  and  the 
potential  difference  between  a  and  x  is  70—  Fa;=Q(l/r0  —  1/r*) 


+  (? 


FIG.  272. 


This  result  does  not  depend  upon  the  path  between  a  and  a; 
(§63).  In  the  case  of  a  number  of  charges  Q',  Q,",  Q'",  etc., 
the  potential  at  a  point  a  is  the  sum  of  the  potentials  due  to  each 
charge,  that  is 


7'"  +  ,  etc.,  =QV^  +  Q^r'^+Q"'/r'''a  +  ,  etc. 

When  the  medium  is  not  air,  the  right-hand  side  of  each  of 
the  above  equations  must  be  multiplied  by  the  proper  value  of 
1/KfoT  that  medium  (§401). 

The  proof  of  the  above  by  calculus  is  very  simple.  We  have  the  differ- 
ential of  work  dW**Fdr=>Qlr*dr.  Integrating  between  the  limits  r0 
and  rn,  we  get  the  total  work,  or  70  —  yw=- 


firV-     (NWA- 

Jr.  Jr."  V* 


354 


ELECTRICITY  AND  MAGNETISM 


407.  Electrometers. — An  instrument  for  measuring  difference 
of  electrical  potential  by  means  of  electrostatic  force,  is  called  an 
electrometer.  In  these  instruments  there  is  a  movable  part — 
a  charged  needle  or  a  disk — which  is 
acted  on  by  an  electric  field  produced 
by  the  difference  of  potential  to  be 
measured.  The  common  forms  of  elec- 
trometers are  the  quadrant,  the  disk 
or  absolute  electrometer,  the  single 
and  the  multi-cellular  "electrostatic 
voltmeters."  All  of  these  were  first 
developed  by  Lord  Kelvin. 


Fio.  273o. — Quadrants. 


FIG.  2736. — Quadrant  Electrometer. 


The  quadrant  electrometer  (Fig.  2736)  consists  of  a  light  needle 
made  of  sheet  aluminum  or  of  silvered  paper,  which  is  suspended 
by  a  fine  metal  strip,  or  by  a  quartz  fiber,  inside  of  a  shallow 
circular  metal  box.  This  metal  box  is  divided  into  four  quadrants 
which  are  mounted  on  insulating  columns,  preferably  of  amber. 
The  diagonally  opposite  quadrants  are  connected  by  wires.  The 
needle  is  free  to  move  in  a  horizontal  plane,  and,  in  its  position  of 
equilibrium,  the  needle  hangs  along  the  line  of  separation  of  the 
two  pairs  of  quadrants.  Any  deflection  of  the  needle  can  be 
read  by  the  movement  of  a  beam  of  light  reflected  from  a  small 
mirror  attached  to  the  suspended  needle.  The  needle  is  charged 
to  a  high  positive  potential,  generally  by  joining  it  through  its 
suspension  to  a  high  potential  battery.  So  long  as  the  two 
pairs  of  quadrants  are  at  the  same  potential,  there  is  no  deflec- 
tion of  the  needle.  But,  if  the  pair  of  quadrants  A  and  D  are  at 
a  higher  potential  than  the  quadrants  B  and  C,  there  will  be  a 
couple  deflecting  the  needle  toward  the  quadrants  at  the  lower 
potential.  This  couple  is  balanced  by  the  torsion  of  the  suspen- 
sion, and  it  can  be  shown  that  for  small  angles,  the  difference  of 


ELECTROSTATICS 


355 


potential  is  proportional  to  the  angle  of  deflection.  The  quad- 
rant electrometer  is  very  sensitive,  a  common  sensitiveness  for  the 
Dolezalek  form  (Fig.  2736)  being  a  deflection  of  1  mm.  at  1 
meter  scale  distance  for  a  difference  of  potential  of  .002  volt. 


Fio.  274. — Electrostatic  Voltmeter. 


Fia.  275. — Braun  Electrometer. 


The  quadrant  electrometer  can  be  used  without  charging  the  needle 
independently.  The  needle  is  then  joined  to  one  pair  of  quadrants,  say 
A  and  D,  so  that  the  only  charges  are  those  due  to  the  difference  of  potential 
to  be  measured.  This  arrangement  is  called  the  idiostatic,  while  the  previous 
arrangement  is  called  the  heterostatic.  The  idiostatic  arrangement  is 
adapted  for  measuring  larger  differences  of  potential.  In  the  "vertica- 
electrostatic  voltmeter"  (Fig.  274)  there  is  a  single  pair  of  vertical  quadl 
rants,  and  the  needle  is  an  aluminum  vane  balanced  on  knife  edges,  the  dif- 

erence  of  potential  between  the  quadrants  and  the  needle  is  indicated 
by  the  tilting  of  the  needle  as  shown 
by  the  pointer  and  scale.  Differ- 
ences of  potential  of  from  1000  to 
20,000  volts  can  be  measured  with 
this  instrument. 

In   the   "multi-cellular  electro- 
static voltmeter,"  the  needle  consists  of  a  series  of  parallel  vanes,  and 

wings  horizontally  between  a   corresponding    series  of  fixed  plates  or 
quadrants.     Increasing  the  number  of  vanes  and  quadrants,  increases  the 

ensitiveness,   so  that  multi-cellular  voltmeters  reading  as  low  as  10  volts 
are  listed  by  the  makers. 

The  Braun  electrometer  (Fig.  275)  has  a  light  needle  that  is  pivoted  on 
a  horizontal  axis,  and  the  action  is  similar  to  that  of  the  modified  gold-leaf 
electroscope  mentioned  in  §395. 

In  the  disk  electrometers  there  are  two  parallel  plates,  A  and  B, 
charged  to  the  potentials  V*  and  Vi>.  Part  of  the  upper  plate,  the  disk 


Fio.  276. 


356 


ELECTRICITY  AND  MAGNETISM 


S,  is  movable  and  hung  on  a  balance  arm  as  in  Fig.  276  (or  it  may  hang 
by  a  calibrated  spring),  so  that  the  force  pulling  it  toward  the  plate  B 
can  be  counterbalanced  and  thus  measured.  The  outer  part  of  the  plate 
A  is  called  the  "guard  ring,"  and  serves  the  purpose  of  making  the 
electric  field  uniform  opposite  the  movable  disk  S,  as  indicated  by  the 
parallel  lines  of  force.  It  can  be  shown  that  in  this  case,  the  difference 
of  potential  is  given  by  the  formula 


where  d  is  the  distance  in  centimeters  between  the  plates,  F  is  the  force 
in  dynes  acting  on  S,  and  S  the  area  in  square  centimeters  of  the  disk. 
The  difference  of  potential  (Va  —  Vb)  is  accordingly  given  in  absolute 
c.g.s.  units  and  so  this  is  an  "  absolute  electrometer." 

It  is  now  seen  that  the  deflection  of  the  gold-leaf  electroscope 
is  due  to  the  difference  of  electrical  potential  between  the  gold- 
leaf  and  the  metal  case  (Fig.  266,  §399),  since  the  lines  of  force 
connect  the  leaf  and  the  case.  The  metal  case  is  ordinarily 
connected  with  the  earth  and  so  is  at  zero  potential.  For  small 
deflections,  the  difference  of  potential  (V^-Vo)  is  proportional  to 
the  deflections. 

Static  Electrical  Machines 

408.  The  Electrophorus.  —  The  electrophorus  is  an  instrument 
devised  by  Alexander  Volta  in  1790,  to  multiply  electric  charges 


Fid.  277. 

by  electrostatic  induction.  It  consists  of  a  resin  plate,  A,  on  a 
metal  plate  or  "sole,"  £,  and  a  metal  disk,  B,  with  an  insulating 
handle,  H,  at  its  center.  The  resin  is  charged  negatively  by 
friction.  The  metal  base  then  has  a  "bound"  positive  charge 
which  helps  to  hold  the  negative  charge  on  the  resin.  Bring 
the  metal  disk  B  near  and  opposite  the  resin  plate.  There  is 
induced  in  B  a  free  negative  charge  and  a  bound  positive  charge. 


ELECTROSTATICS  357 

The  free  charge  is  removed  by  connecting  B  with  the  earth  for  an 
instant.  If  the  plate  B  is  then  moved  away  from  the  resin  plate 
A,  the  positive  charge  is  made  a  free  charge  and  can  be  transferred 
to  another  conductor,  such  as  an  insulated  sphere.  This  process 
can  be  repeated,  the  plate  B  being  charged  each  time  without 
decreasing  the  charge  on  the  resin  plate. 

As  described  above,  the  plate  B  is  brought  "near"  A,  but  in 
practice,  the  plate  B  is  actually  rested  on  A.  The  points  of  con- 
tact are,  however,  only  comparatively  few,  so  that  little  of  the 
negative  charge  of  the  resin  is  removed  to  B,  while  the  charge 
induced  on  B  at  the  shorter  distance  is  increased.  Instead  of 
a  resin  plate,  a  plate  of  sulphur,  hard  rubber,  shellac,  or  other 
insulator  can  be  used. 

The  positive  charge  on  the  plate  B,  represents  energy.  The 
source  of  this  energy  is  in  the  work  done  in  lifting  the  plate  B 
after  the  free  negative  charge  q  has  escaped.  Work  is  done  in 
pulling  apart  the  charge  of  the  plate  and  the  disk,  that  is,  in 
drawing  out  the  lines  of  force  connecting  the  plus  and  minus 
charges.  The  energy  lies  in  this  tension  in  the  lines  of  force 
in  the  dielectric. 

409.  Electrostatic  Induction  Machines. — The  earliest  machines 
for  producing  electrical  charges 
by  rotation  were  friction  ma- 
chines.    Thus  a  glass  plate  was 
electrified   by  rotating   it  be- 
tween  suitable    rubbers,    and 
the  charge  removed  by  metal 
brushes  or  pointed  conductors.     A 
Such  friction  machines  are  sel-     + 
dom  found  now  outside  of  mu- 
seums, and  have  only  historical 
interest.      They     have     been 
superseded  by  the  electrostatic  Flo  278 

induction  machines.     The  two 

most  common  of  these  machines  are  the  Toepler-Holtz,  and  the 
Wimshurst  machines. 

The  Toepler-Holtz  or  Voss  machine  has  two  vertical  glass  or 
vulcanite  disks,  of  which  one  is  fixed  and  the  other  rotates 
about  a  horizontal  axis  normal  to  its  center.  The  positions  of 


358 


ELECTRICITY  AND  MAGNETISM 


the  various  parts  can  be  seen  in  Fig.  278  (due  to  Prof.  S.  P. 
Thompson),  in  which  circles  are  used  to  represent  the  two  disks; 
the  farther  disk  which  is  fixed,  is  represented  by  the  outer  circle, 
and  the  nearer  and  rotating  disk  is  representd  by  the  inner 
circle.  On  the  far  side  of  the  fixed  disk  are  the  combined  paper 
and  tin-foil  "inductors,"  A  and  B.  On  the  near  face  of  the 
rotating  disk,  there  are  six  tin-foil  disks  p,q,r,  etc.,  called 
"  carriers."  These  are  spaced  at  equal  distances  around  the  disk. 
A  "  neutralizing  rod,"  n^n^  reaches  across  the  front  face  of  the 
rotating  disk,  and  by  small  brushes  connects  the  two  opposite 
carriers  as  they  pass  under  the  rod.  On  each  inductor  there  is 
a  metal  brush  arranged  so  as  to  make  contact  with  each  carrier 
as  it  passes.  Collecting  "combs"  (§397),  Clf  and  C2,  with  dis- 
charge rods  and  balls,  D,  are  as  shown,  arranged  in  front  of 
the  rotating  plate. 

The  action  of  the  machine  is  as  follows:  One  of  the  inductors, 
say   A  acquires  by  friction  a  small  initial  positive  charge.     This 

induces  on  the  carrier  at  p, 
a  bound  negative  charge  and 
a  free  positive  charge.  The 
positive  charge  escapes  along 
the  neutralizing  rod.  The 
negative  charge  becomes  free 
as  the  carrier  passes  to  the 
position  q.  Here  it  shares 
its  negative  charge  with  the 
inductor  B.  At  r  it  loses  its 
negative  charge  to  the  comb 
C,  and  thus  the  ball  Bt  is 
charged  negatively.  At  s  the 
carrier  has  a  bound  positive 
charge  by  induction  from  the 
negative  inductor  B,  the  free 
negative  charge  escaping  along  the  neutralizing  rod.  At  t  the 
carrier  shares  its  charge  with  the  inductor  A ,  and  at  u  it  loses 
its  positive  charge  to  the  comb  Cr  The  ball  B1  is  thus  the  posi- 
tive terminal.  The  difference  of  potential  between  B1  and  Bt 
is  thus  increased  until  a  spark  discharge  takes  place.  A  Leyden 
jar  is  usually  connected  to  each  terminal,  the  effect  of  which  is 


ELECTROSTATICS  359 

to  increase  the  quantity  of  discharge  for  a  given  potential  dif- 
ference (see  condensers  §410).  In  a  new  machine,  made  by 
Wehrsen,  the  rotating  plate  of  ebonite  is  triple  and  contains 
embedded  metal  sectors  connected  with  the  carriers,  thus  greatly 
increasing  the  output. 

In  the  Wimshurst  machine  (Fig.  279)  there  are  two  parallel 
glass  disks,  geared  to  rotate  in  opposite  directions.  On  each  disk 
is  a  large  number  of  tin-foil  sectors,  and  each  sector  serves  in 
turn  as  inductor  and  carrier.  The  neutralizing  rods  and  combs 
are  symmetrical  on  the  two  sides.  The  action  of  the  machine 
is  similar  to  that  of  the  previous  machine  and  can  be  followed 
from  the  +  and  —  signs  on  the  figures. 

410.  Electrical  Capacity. — If  two  metal  balls  of  different  sizes 
be  put  in  contact  and  charged,  they  will  be  at  the  same  electrical 
potential,  but  they  will  not  have  the  same  electrical  charges. 
This  can  be  shown  by  hanging  each  ball  separately  in  a  metal 
cup  on  the  plate  of  a  gold-leaf  electroscope  and  noting  the  diverg- 
ence of  the  leaves  (Fig.  265,  §399).  The  fact  that  it  takes  more 
electricity  to  raise  the  potential  of  A  a  certain  amount  than  to 
raise  the  potential  of  B  the  same  amount,  we  describe  by  saying 
that  the  "electrical  capacity"  of  A  is  greater  than  the  electrical 
capacity  of  B. 

Electrical  capacity  may  be  illustrated  by  the  capacity  of  a 
tank  for  gas.  The  mass  of  gas  (neglecting  temperature  changes) 
depends  upon  two  things,  (a)  the  dimensions  of  the  tank,  and 
(6)  the  pressure  of  the  gas.  The  mass,  M ,  of  the  gas  equals  the 
pressure,  P,  of  the  gas,  multiplied  by  K,  the  mass  of  the  gas  in 
the  tank  at  unit  pressure,  or  M  =  KP.  Similarly,  the  electrical 
charge  Q  on  a  conductor  is  equal  to  the  electrical  potential  V, 
multiplied  by  C,  the  electrical  charge  on  the  conductor  at  unit 
potential,  or  Q  =  CV,  assuming  that  V  is  due  entirely  to  Q,  that  is, 
that  the  conductor  is  not  in  a  field  due  to  other  charges.  In 
this  statement,  C  is  called  the  electrical  capacity  of  the  con- 
ductor; it  is  the  quantity  of  electricity  required  to  raise  the  poten- 
tial of  the  conductor  by  unit  amount. 

We  can  thus  compare  the  electrical  capacities  of  two  con- 
ductors A  and  B,  either,  (1)  by  comparing  the  charges  required 
to  raise  them  to  the  same  potential,  or  (2)  by  comparing  the 
potentials  to  which  equal  charges  raise  the  two  conductors.  In 


360 


ELECTRICITY  AND  MAGNETISM 


the  latter  case,  the  greater  the  capacity,  the  less  a  given  electrical 
charge  would  raise  the  potential. 

It  can  be  shown  that  the  electrical  capacity  of  a  conductor 
depends  not  only  (a)  upon  its  size,  but  also  (b)  upon  its 
shape,  and  (c)  upon  the  position  of  neigh- 
boring conductors,  and  (d)  upon  the  sur- 
rounding insulating  medium  or  dielectric. 
That  the  capacity  of  a  conductor  depends 
upon  the  shape  can  be  demonstrated  by 
the  apparatus  shown  in  Fig.  280.  The 
conductor  is  connected  by  a  wire  with  an 
electroscope.  A  charge  Q  is  given  to  it, 
and  this  raises  the  potential  of  the  system 
to  V,  as  indicated  by  the  divergence  of  the 
electroscope.  The  conductor,  being  made 
of  a  series  of  cups,  can  now  be  drawn  out, 
thus  changing  the  shape  of  the  conductor. 
The  electroscope  leaves  converge,  indicat- 
ing a  lowering  of  potential,  although  the 
Q  charge  on  the  conductor  is  not  changed. 
Upon  restoring  the  shape  of  the  conductor 
the  potential  is  restored. 

The  fact  that  the  capacity  depends  upon  the  position  of  neigh- 
boring conductors  is  shown  by  the  apparatus  illustrated  in  Fig. 
281.  A,  an  insulated  metal  B  A 

disk,  is  connected  by  a  wire 
with  the  electroscope  E.  The 
divergence  of  E  indicates  as 
before  the  potential,  V,  caused 
by  a  charge  Q.  Now  bring  up 
the  earthed  disk  B,  and  the 
potential  V  is  lowered  as  in- 
dicated by  the  convergence 
of  the  electroscope.  That  is, 
to  raise  A  to  the  potential  V 
requires  a  greater  charge  if  B  is  nearer  A;  in  other  words  the 
electrical  capacity  of  A  is  increased  by  the  presence  of  the  con- 
ductor B.  If  a  plate  of  hard  rubber,  sulphur,  or  glass  be  now 
put  between  A  and  B,  the  electroscope  converges;  that  is,  the 


Fio.  280. 


I 

Earth  ^^ 

4 

4- 

4- 
4- 

J                              f 

1 

+/  \4 
V    \  + 

Fio.  281. 


ELECTROSTATICS 


361 


potential  falls  while  the  charges  remain  the  same.  The  elec- 
trical capacity  thus  depends  upon  the  intervening  medium. 
We  thus  see  that  the  electrical  capacity  of  a  conductor  depends 
upon  (a)  the  size,  (6)  the  shape  of  the  conductor,  (c)  the  position 
of  neighboring  conductors,  and  (d)  the  intervening  dielectrics. 
The  arrangement  shown  in  Fig.  282,  consisting  of  two  conductors 
separated  by  a  dielectric,  thus  serves  to  increase  the  electrical 
capacity  of  the  insulated  conductor,  and  is  called  an  electric 
condenser. 

That  the  electrical  capacity  of  the  conductor  A  is  increased 
by  the  presence  of  the  conductor  B  follows  from  the  definition 
of  electrical  potential.     The  potential  of  A  is 
equal  to  the  work  required  to  bring  a  test     —Q     +Q 
unit  charge  from  the  earth  to  the  body  A 
against  the  force  of  the  electrical  field  (Fig. 
282).      But  if  we  have  not  only  the  charge 
+  Q,   but  also  the  induced  opposite   charge 
—  Q,  the  force  acting  on  the  test  unit  is  less; 
that  is,  the  work  is  less,  or  the  potential  of  A 
is  lowered.     Hence  the  charge  to  raise  A  to 
a  given  potential  must  be  greater;  that  is,  the 
electrical  capacity  of  A  is  increased. 

411.  Condensers. — The  arrangement  of  two  conductors  sepa- 
rated by  an  insulator  or  dielectric  is  that  found  in  the  Leyden 
jar  (Fig.  283).  This  consists  of  a  glass  jar,  coated  inside  and 


O 


B         A 
Fio.  282. 


Fio.  283. 


Fia.  284. 


outside  for  about  two-thirds  of  its  height  by  tin-foil.     Connec- 
tion with  the  inside  is  made  by  a  supported  brass  rod  terminat- 


362  ELECTRICITY  AND  MAGNETISM 

ing  in  a  ball.     When  the  inside  coating  A  is  charged,  an  opposite 

r'^^^^^^^m^^  "  bound  "  charge  is  induced  on 

s:^^/^,^^^  the  outside  coating  B,  and  the 

-rfy   \^     wwMf/  y///////////////////,      >^.  t  ° 

X,  >//////////////////////////////////^///^  ^/P-      *ree    induced  charge  escapes 

'/P/W///J///////////////////////////////////////////////S/     -J  ,  ,  -       ,  . 

to  the  earth.     If  the  inner  and 
outer  coatings  are  connected, 

there  is  a  spark  and  an  electric  discharge  which  is  often  very 
violent.  In  discharging  a  Leyden  jar,  it  is  safe  and  convenient 
to  use  discharge  tongs  with  an  insulating  handle  (Fig.  284). 

Very  compact  condensers  are 
made  by  piling  up  sheets  of  tin- 
foil separated  by  sheets  of  mica 
or  of  paraffined  paper  (Fig.  285). 
The  alternate  tin-foil  sheets  are 
joined  and  thus  a  great  capacity 
is  secured  in  a  small  space. 
Mica  condensers  are  used  in  the 
laboratory  for  standards  in  test- 
ing. Fig.  286  shows  a  common 
form.  The  condensers  in  the 
bases  of  induction  coils  (§511) 
are  generally  paraffined-paper  condensers. 

412.  Residual  Discharge. — A  succession  of  discharges  can  usu- 
ally be  obtained  from  a  charged  Leyden  jar  or  other  condenser. 
Thus,  when  the  inner  and  outer  coatings  are  connected,  there 
is  a  brilliant  spark,  and,  if  they  are  connected  a  half  minute  later, 
still  another  spark  discharge  takes  place;  and  this  may  often  be 
repeated  a  half  dozen  times,  each  successive  discharge  being  less 
than  the  preceding.  These  later  discharges  are  called  residual 
discharges.  The  number  and  magnitude  of  the  residual  dis- 
charges differ  greatly  for  different  condensers,  depending  upon 
the  kind  and  thickness  of  the  dielectric.  Air  condensers  show 
no  residual  discharges. 

The  residual  discharges  are  explained  as  due  to  the  "absorp- 
tion "  of  the  charges  by  the  dielectric,  and  the  gradual  escape  of 
these  charges.  The  "absorption"  is,  however,  probably  a  state 
of  strain  with  associated  stresses  in  the  dielectric.  Just  as  rubber 
recovers  gradually  from  distortion  (§179),  so  the  dielectric  takes 


ELECTROSTATICS  363 

time  to  recover.     Homogeneous  dielectrics  such  as  gases  and 
quartz  show  no  residual  effects. 

The  above  assumes  that  the  energy  of  the  charged  Leyden  jar  is 
to  be  found  in  the  glass,  or  other  dielectric.  A  very  beautiful 
experiment  made  by  Benjamin  Franklin  as  early  as  1748, 
showed  that,  in  the  Leyden  jar,  "the  whole  force  of  the  bottle 
and  power  of  giving  a  shock  is  in  the  glass  itself;  the  non-electrics 
(conductors)  in  contact  with  two  surfaces  serving  only  to  give 
and  receive  to  and  from  the  several  parts  of  glass;  that  is,  to 
give  on  one  side  and  take  away  on  the  other.17  Franklin  used  a 
glass  jar  coated  on  the  outside  with  lead  foil  and  having  water 
on  the  inside  for  the  inner  conductor.  Connection  with  the 
water  was  made  by  a  wire  supported  by  the  cork  stopper  of  the 


FIG.  287. 

bottle.  The  jar  or  bottle  was  charged  from  an  electrical  machine. 
The  water  was  then  poured  out  and  found  to  be  uncharged. 
Fresh  water  was  poured  into  the  bottle,  and  a  bright  spark  was 
obtained  by  discharge  of  the  jar.  A  common  apparatus  for 
Franklin's  experiment,  known  as  the  separable  Leyden  jar,  is 
shown  in  Fig.  287.  It  consists  of  a  glass  cup,  Gt  hi  which  the 
metal  cone  A^  fits  as  inner  conductor,  and  the  metal  cup  Blt  in 
which  the  glass  cup  G  fits.  A  is  charged,  and  the  usual  discharge 
takes  place  upon  connecting  A1  with  Br  If  A t  is  again  charged, 
and  then  lifted  out  with  a  rubber  handle,  and  B  is  removed,  it 
is  found  that  neither  Al  nor  Bl  is  charged.  If  now  the  jar  be  built 
up  in  the  reverse  order  with  the  same  glass  cup  G,  but  with  a 
second  set  of  conductors,  A2  and  Bt,  a  bright  spark  discharge  is 


364 


ELECTRICITY  AND  MAGNETISM 


obtained  from  the  Leyden  jar.     From  this  we  conclude  that  the 
essential  part  is  the  dielectric. 

413.  Dielectric  Properties.  Specific  Inductive  Capacity.— The 
first  person  to  publish  a  systematic  study  of  the  dielectrics  in 
condensers  was  Michael  Faraday.  Faraday  used  two  similar 
spherical  condensers,  each  consisting  of  a  brass 
sphere,  A,  (Fig.  288),  suspended  by  an  insulating 
support  at  the  center  of  a  hollow  spherical  shell, 
B.  The  shell  was  made  of  two  flanged  hemi- 
spheres which  could  be  separated  so  that  differ- 
ent dielectrics  could  be  introduced.  One  of  the 
condensers  with  air  for  dielectric  was  given  a 
charge  Q.  Its  potential,  7,  was  then  7=Q/C. 
If  this  condenser  was  then  joined  to  the  second 
condenser  with  air  also  as  dielectric,  the  charge 
Q  was  shared  equally  between  the  two,  and  the 
potential  7  was  halved.  But  if  the  second 
condenser  had  another  dielectric,  say  sulphur, 
the  original  potential  was  reduced  to  7',  which 
was  less  than  half  of  7.  Let  C"  be  the  capacity 
of  the  second  condenser.  Then  (Cf  +  C)7' =Q 
and  C7=Q,  and  hence  C"/C  =  (7-  7')/7'.  Faraday  found 
that  his  condenser  with  sulphur  forming  part  of  the  dielectric 
had  a  capacity  1.6  times  the  capacity  of  a  similar  condenser 
with  air  alone  as  dielectric.  Hence,  the  "inductive  action" 
across  sulphur  must  be  greater  than  across  air. 

Faraday  in  this  way  discovered  that  "different  dielectric 
bodies  possess  an  influence  over  the  degree  of  induction  which 
takes  place  through  them."  He  described  this  by  saying  that 
these  dielectrics  had  different  "specific  inductive  capacities." 
The  specific  inductive  capacity  or  "dielectric  constant"  of  a  sub- 
stance, as  it  is  called  more  frequently  now,  is  the  ratio  of  the  capacity 
of  a  condenser  with  the  given  substance  as  dielectric  to  the  capacity 
of  the  same  condenser  with  air  as  dielectric.  The  following  is  a 
list  of  the  dielectric  constants  of  some  common  dielectrics: 


FIG.  288. 


Paraffine  wax 2.0   to  2.3 

Petroleum 2 . 07 

Hard  rubber 2.0    to  3.1 

Sulphur 2.2    to  4. 


Mica 6.     to  8. 

Glass 6. 6  to  9. 9 

Distilled  water 75.+ 

Alcohol . .  .25.+ 


ELECTROSTATICS  365 

These  are  the  same  constants  as  appear  in  Coulomb's  law  of 
electrical  force  (§401),  and  one  method  of  obtaining  these 
constants  is  to  measure  the  force  between  charged  conductors 
in  the  different  dielectrics.  The  reason  tor  the  connection  is 
readily  seen  from  the  remark  at  the  end  of  §406. 

414.  Units  of  Capacity. — A  conductor  or  condenser  has  unit 
capacity  when   unit  quantity  of  electricity  raises  it  to   unit 
potential;  that  is,  C  is  unity  when  Q  and  V  are  each  unity.     If 
Q  and  V  are  expressed  in  c.g.s.  electrostatic  units  of  quantity  and 
potential  (§405),  the  above  is  the  definition  of  the  c.g.s.  electro- 
static unit  of  capacity. 

If  we  use  the  coulomb  and  volt  as  the  units  of  quantity  and 
potential  (§441),  the  corresponding  unit  of  capacity  is  the  farad. 
The  farad  is  the  electrical  capacity  of  a  conductor  which  requires 
a  coulomb  to  raise  its  potential  to  one  volt.  From  the  relations 
C  =  Q/V;  1  coulomb  =3  XlO9  c.g.s.  electrostatic  units;  and 
1  volt  =l/3XlO~a  c.g.s.  electrostatic  units;  we  get  1  farad 
=  9X(10)U  c.g.s.  electrostatic  units  of  capacity.  The  micro- 
farad, the  millionth  of  a  farad,  is  the  ordinary  practical  unit  of 
capacity.  From  the  above  we  see  that  a  microfarad  =  9x(10)8 
c.g.s.  e.s.  units. 

415.  Capacity  Calculations. — The  capacity  of  certain  forms 
of  conductors  and  condensers  can'  be  calculated  when  the  dimen- 
sions of  the  parts,  and  the  dielectric  constant  of  the  insulator  are 
known.     We  give  here  the  formulae  for  the  capacity  C  of  a  sphere 
and  of  three  forms  of  condensers.     The  dielectric  constant  K  is 
unity  if  air  is  the  insulating  medium. 

For  an  isolated  sphere,  with  radius  r,  in  air, 


For  at  all  points  inside  the  sphere  the  potential  is  the  same  and 
equal  to  that  at  the  surface  of  the  sphere.  Since  all  parts  of  the 
charge  on  the  surface  of  the  sphere  are  at  the  same  distance  from 
the  center,  the  potential  at  the  center  is  V  =  Q/r  or  Q/F  =  r. 
Hence  the  capacity  of  the  sphere  is  equal  to  r. 

For  a  sphere  of  radius  r  surrounded  by  a  concentric  spherical 
shell  of  internal  radius  r'  (Fig.  289). 

iwiSi 


366  ELECTRICITY  AND  MAGNETISM 

For,  if  the  charge  on  the  inner,  sphere  is  Q,  that  on  the  outer 
sphere  is  -Q.  The  outer  sphere,  being  connected  to  the  earth,  is 
at  zero  potential.  The  potential  at  any  point  on  the  surface  of 
the  inner  sphere  due  to  the  charge  on  the  outer  sphere  is  -Q// 
when  the  dielectric  is  air  and  that  due  to  the  charge  on  the  inner 
sphere  is  Q/r.  Hence  the  potential  of  the  inner  sphere  is  V  = 
Q(l/r-l/r')etc. 


Fio.  289.  Fia.  290. 

For  a  cylinder  of  radius  r  surrounded  by  a  concentric  cylindrical 
shell  of  internal  radius  rf  (Fig.  290),  and  of  length  I,  large  com- 
pared to  r  and  / 

I 


For  a  pair  of  equal  parallel  plates  of  area  A,  a  relatively  small 
distance,  d,  apart 


The  proofs  of  these  formulae  can  be  found  in  the  more  extended 
treatises  on  electricity. 

416.  Energy  of  a  Charged  Conductor.  —  The  discharge  spark 
with  its  accompanying  light  and  heat  is  sufficient  evidence  that  a 
charged  conductor  or  condenser  has  energy.  An  expression  for 
this  energy  is  most  easily  found  by  calculating  the  work  re- 
quired to  charge  the  body.  Let  Q  equal  the  charge  required  to 
raise  the  body  to  the  potential  V.  The  potential  of  a  body  has 
been  defined  as  the  work  required  to  bring  a  positive  test  unit 
from  infinity  to  the  body.  The  definition  assumes  that  the 
potential  of  the  conductor  is  not  appreciably  changed  by  the 


ELECTROSTATICS 


367 


addition  of  the  test  unit.  Here,  however,  we,  have  to  determine 
the  work  necessary  to  bring  a  charge  Q  up  to  a  body  which  is 
initially  at  zero  potential,  but  which  is  v 

raise  1  to  the  final  potential  V  by  the 
charge  Q.  We  can  think  of  the  charge 
as  being  brought  in  a  very  large  num- 
ber of  small  fractional  charges  Q/n. 
The  potential  of  the  body  will  be 
raised  equal  amounts  as  each  frac- 
tional charge  is  added,  until  the  final 
potential  V  is  reached.  The  average 
potential  during  the  charge  is  thus 
F/2,  and  hence  the  total  work  equals  the  product  of  the  total, 
charge  Q  and  the  average  potential,  and  so  the  energy  E  =  QV/2. 
The  process  can  also  be  represented  graphically  (Fig.  291)  and 
the  result  obtained  from  the  area  of  a  triangle  as  in  the  similar 
cases  considered  in  §§27,  56.  Since  Q=CV  (§410),  we  can 
write  this  in  the  form,  E  =  CV*/2. 


Q 

n 
FIQ.  291. 


Q 


• 

Fio.  292. 

417.  Condenser?  "In  Parallel"  and  "In  Series." — Condensers 
can  be  joined  together  "in  parallel"  or  "in  series."  When  joined 
in  parallel,  all  the  positive  coatings  are  connected  to  form  one 
terminal  and  side  of  the  battery  of  condensers,  and  all  the  nega- 
tive coatings  are  connected  to  form  the  other  terminal  and  side 
(Fig.  292).  The  resultant  capacity  is  that  of  a  single  large  con- 
denser, with  coatings  equal  to  the  sum  of  the  coatings  of  the  in- 
dividual condensers,  or  C  =  Cl  +  C2  +  C8  + ,  etc 

Condensers  are  joined,  "in  series"  (or  "in  cascade")  when  they 
are  insulated,  and  the  outer  coating  of  the  first  is  connected  to  the 
inner  coating  of  the  second  and  the  outer  coating  of  the  second  is 


368 


ELECTRICITY  AND  MAGNETISM 


joined  to  the  inner  coating  of  the  third,  and  so  on  through  the 
series  of  condensers  (Fig.  293).  To  calculate  the  capacity  of 
the  battery  "in  series,"  let  Vlt  72,  73,  etc.,  be  the  potentials 
and  Cj,  C2,  C8,  etc.,  the  capacities  of  the  condensers.  The 


Fio.  293. 


quantity  Q  on  each  inner  coating  will  in  each  case  equal  the 
quantity  on  the  outer  coating,  and  this  will  equal  the  quantity 
on  the  next  inner  coating  since  they  are  corresponding  induced 
charges.  Thus 


where  C  is  the  resultant    capacity    of    all    the   series.     Hence 


_ 

8 


and 


418.  Quantitative  Use  of  Lines  and  Tubes  of  Force. — In  rep- 
resenting the  state  of  an  electric  field  by  lines  of  force,  only  the 
question  of  the  direction  of  the  force  has  been  considered.  To 
represent  the  magnitude  of  the  force,  we  limit  the  number  of 
lines  of  force  from  a  body,  so  that  the  number  of  lines  is  equal  to 
the  number  of  units  of  positive  charge.  That  is,  if  there  are  q 
units  of  electricity,  there  will  be  q  lines  of  force.  Each  line  is 
then  called  "  a  unit  line,"  or  simply  a  "line  of  force." 

Suppose  the  charge  q  to  be  at  a  point,  and  distant  from  other 
charges.  The  lines  are  then  radial  and  symmetrical  about  q. 
Draw  a  sphere  with  radius  r  and  with  q  at  the  center.  Through 
this  spherical  surface,  there  are  A^  =  ?/47rr3  lines  per  square  centi- 


ELECTROSTATICS  369 


meter.  But  the  force  on  unit  charge  at  adistance  r  is  F 
(§401)  hence  F=4xN/K,  where  N  is  the  number  of  unit  lines 
through  a  square  centimeter  section  taken  normal  to  the  field. 
But  the  force  F  acting  on  unit  charge  at  a  point  is  called  the 
intensity  of  the  field  at  the  point.  The  intensity  of  the  field  at 
a  point  is  thus  equal  to  4n/K  times  the  number  of  unit  lines 
per  square  centimeter  of  normal  section  of  the  field. 

For  the  more  complete  representation  of  the  state  of  an  elec- 
tric field,  we  introduce  the  conception  of  "tubes  of  force."  A 
tube  of  force  is  a  channel  bounded  by  lines  of  force  and  having, 
as  one  end,  the  area  covered  by  a  positive  charge,  and,  as  the 
other  end,  the  area  covered  by  the  corresponding  negative  charge. 
A  "unit  tube"  or  a  "Faraday  tube"  as  Professor  J.  J.  Thomson 
calls  it,  has  unit  positive  charge  at  one  end,  and  unit  negative 
charge  at  the  other  end.  Hence  as  many  unit  tubes  start  from 
a  body  as  the  body  has  units  of  positive  electricity  on  it.  The 
electric  field  is  thus  filled  with  these  tubes  of  force. 

The  terms  "unit  line"  and  "unit  tube"  are  thus  equivalent. 
We  see  that  the  intensity  at  a  point  of  a  field  is  equal  to  4^/K 
times  the  number  of  tubes  of  force  per  square  centimeter  of  sec- 
tion normal  to  the  field,  or  F=4xN/K.  If  there  are  N  tubes  across 
a  square  centimeter,  then  s,  the  cross-section  of  each  tube,  is 
s  =  1  /  N=4n:/FK.  Hence  Fs  —^njK,  or  the  product  of  the  cross- 
section  of  a  tube  and  the  intensity  of  field  is  the  same  at  every  sec- 
tion of  the  tube.  We  thus  see  that  where  the  intensity  decreases 
the  tube  widens  out,  and  where  the  intensity  increases  the  tube 
narrows.  This  suggests  the  flow  of  a  stream,  and  hence  Maxwell 
and  others  use  the  term  "flux"  or  "flow"  of  force,  for  the  quan- 
tity Fs. 

419.  Energy  of  an  Electric  Field.  —  To  calculate  the  energy  in  unit 
volume  of  an  electric  field,  take  the  case  of  two  parallel  charged  plates, 
with  a  difference  of  potential  V  and  a  charge  of  Q.  Q  unit  tubes  extend 
between  the  two  plates.  The  total  energy  is  (VQ/2)  (§416),  and  so  the 
energy  per  tube  is  V/2  ergs.  If  the  tube  has  a  length  L,  the  energy  per 
unit  length  of  tube  is  V/2L.  But  V  =  FL  by  the  definition  of  difference  of 
potential  and  so  the  energy  per  unit  length  of  the  tube  is  F/2.  We  have 
already  seen  that  the  number  of  tubes  across  a  normal  square  centimeter  is 
N  *=FK/4.n  (§418)  But  N  tubes  each  of  unit  length,  occupy  a  cubic  centi- 
meter. Hence  the  energy  per  cubic  centimeter  of  the  medium  is  F*K/8it. 
24 


370  ELECTRICITY  AND  MAGNETISM 

420.  Atmospheric  Electricity.— In  1752  Benjamin  Franklin 
described  the  famous  kite  experiment  by  which  he  "  completely 
demonstrated  the  sameness  of  the  electrical  matter  with  that 
of  lightning."  A  silk  kite  on  which  there  was  a  pointed  wire 
was  raised,  and  it  was  found  that  "  as  soon  as  any  of  the  thunder- 
clouds came  over  the  kite,  the  pointed  wire  drew  the  electric 
fire  from  them,  and  the  kite  with  all  the  twine  became  electri- 
fied." By  this  means  Franklin  got  electric  sparks  and  also  was 
able  to  charge  a  Leyden  "  phial "  from  the  clouds. 

The  next  marked  advance  in  the  study  of  atmospheric  elec- 
tricity was  due  to  the  invention  and  use  of  the  water-drop 
electrograph  and  the  electrometer  by  Kelvin  about  the  middle 
of  the  last  century.  Kelvin  showed  that  the  end  of  a  tube 
from  which  a  stream  of  water  breaks  into  drops  takes  the  electric 
potential  of  the  air  at  the  point.  It  was  found  that  the  potential 
of  the  air  in  dry  weather  is  normally  positive  relative  to  the 
earth  and  increased  with  the  height.  The  potential  gradient 
is  expressed  in  volts  per  meter  rise  in  height.  This  may  be 
several  hundred  volts  per  meter,  but  it  varies  greatly  with  the 
season,  the  time  of  day  and  the  weather  conditions.  It  is  also 
not  always  positive,  for  the  potential  of  the  atmosphere  at  times 
is  negative  relative  to  the  earth,  and  is  very  frequently  so  in 
rainy  weather.  The  causes  of  atmospheric  electricity  are  not 
definitely  determined.  Evaporation,  friction  of  the  clouds,  the 
action  of  ultra-violet  light,  and  of  radio-active  materials  are 
some  of  the  causes  suggested. 

The  electrical  phenomena  of  the  atmosphere  more  commonly 
observed  are  forms  of  lightning  and  the  aurora  borealis.  Light- 
ning is  an  electric  discharge  between  clouds,  or  between  the 
clouds  and  the  earth.  It  takes  the  form  of  forked  lightning, 
sheet  or  "heat"  lightning  and  "ball"  lightning.  The  "ball" 
lightning  is  not  well  understood  and  may  be  due  to  an  optical 
illusion.  Recent  experiments  by  the  resonance  methods  (§542) 
of  electrical  waves  show  that  lightning  discharges  are  oscillatory 
(§541). 

One  of  the  first  applications  of  electrical  science  was  Franklin's 
use  of  points,  "lightning  rods,"  for  the  protection  of  buildings 
against  injury  by  lightning  discharges.  The  protection  from 
lightning  rods  is  probably  greater  in  the  way  of  silently  dis- 


ELECTROKINETICS  371 

charging  the  surrounding  atmosphere,  rather  than  in  conducting 
away  disruptive  discharges.  The  best  protection  against  light- 
ning is  a  metallic  net  work  covering  the  building  more  or  less 
completely  and  having  a  good  connection  to  moist  earth. 

The  aurora  borealis  or  "  northern  lights,"  is  an  electrical  dis- 
charge in  the  upper  atmosphere,  and  is  most  frequently  seen 
towards  the  polar  regions.  It  is  thought  to  be  analogous  to  the 
electrical  discharges  in  vacuum  tubes. 

ELECTROKINETICS 

421.  The  Electric  Current. — If  two   conductors   A   and   £, 
charged  to  different  electrical  potentials,  be  connected  by  a  long 

>  thin  wire,  there  will  be  a  flow  of  electricity,  that  is,  an  electric  cur- 
rent, through  the  wire.  The  direction  of  this  current  is  defined  as 
being  from  the  higher  to  the  lower  potential.  The  current  will 
continue  so  long  as  there  is  a  difference  of  potential  between  the 
ends  of  the  wire.  In  the  case  of  electrostatic  charges,  such  as 
those  of  the  Leyden  jar,  the  potentials  are  equalized  in  a  very 
small  fraction  of  a  second,  that  is,  the  currents  are  momentary. 
(We  shall  see  later  in  §§541  and  542  that  under  frequent  condi- 
tions, especially  if  the  connecting  wire  is  short  and  thick,  they  are 
also  oscillatory.)  To  get  a  continued  current  in  a  wire,  the 
difference  of  potential  must  be  kept  up.  The  potentials  produced 
by  electrostatic  machines  are  very  high  and  the  electric  quantities 
separated  are  small,  so  that  the  currents  from  such  machines  are 
small,  momentary  and  intermittent.  For  producing  and  main- 
taining continued  electric  currents,  voltaic  cells,  thermo-couples 
and  dynamo-electric  machines  are  used. 

These  will  be  described  later,  but  it  will  be  convenient  to  state 
at  this  stage  the  principle  of  the  voltaic  cell,  since  it  was  the  first 
device  discovered  for  obtaining  continued  currents  and  voltaic 
cells  of  some  form  are  usually  employed  for  most  of  the  experi- 
ments which  we  shall  presently  describe. 

422.  The  Voltaic  Cell.— When  two  dissimilar  conductors,  A 
and  C  (Fig.  294a)   are  immersed  in  a  liquid,  B,  which   acts 
chemically  on  at  least  one  of  them,  and  the  parts  out  of  the  liquid 
are  connected  by  a  wire,  D,  a  current  of  electricity  flows  in  the 
wire,  heating  it  and  producing  other  effects  to   be   described 


372 


ELECTRICITY  AND  MAGNETISM 


presently.  If  the  wire  D  be  cut  and  its  free  ends  be  joined  to  a 
sufficiently  sensitive  electrometer  (§407)  the  latter  will  show  by 
the  deflection  of  its  needle  that  the  ends  of  the  wire  are  at  differ- 
ent potentials.  Volta,  to  whom  we  owe  the  above  discoveries, 
accounted  for  this  difference  of  potential  by  assuming  that  there 
is  an  abrupt  difference  of  potential  set  up  at  the  contact  of  each 
pair  of  dissimilar  conductors  in  the  circuit.  This  view  has  been 
generally  accepted  but  much  difference  of  opinion  has  existed  as 


Fio.  294a. 


Fia. 


to  the  relative  magnitudes  of  these  differences  of  potential  at  the 
various  contacts,  D  with  A,  A  with  B,  B  with  C,  and  C  with  D. 
It  will  not  be  necessary  to  consider  this  controversy  further  at 
present  (see  §475). 

423.  Electromotive  Force. — Assuming  that  there  are  such  con- 
tact differences  of  potential,  let  us  denote  the  rise  of  potential 
from  D  to  A  (positive  or  negative)  by  VDA  and  so  on  for  the  other 
contacts  in  the  circuit.  Then  the  whole  difference  of  potential 
of  the  ends  of  D  connected  to  the  electrometer  and  measured  by 
it  is  VDA  +  VAB  +  VBC  +  VCD.  This,  by  the  definition  of  poten- 
tial difference  (§402) ,  is  the  work  that  would  be  done  in  taking 
a  unit  quantity  of  electricity  from  one  free  end  of  D  to  the  other 
along  the  line  of  the  conductors  in  the  circuit.  When  the  ends 
of  D  are  joined  and  a  current  flows,  we  may  regard  the  current 
as  being  due  to  the  sum  of  the  steps  of  potential  at  the  contacts 
and  this  sum  is  accordingly  called  the  electromotive  force,  Et 
acting  in  the  circuit.  It  is  evidently  also  equal  to  the  work  thai 
would  be  done  in  taking  a  unit  quantity  once  around  the  circuit. 
The  latter  is  the  general  measure  of  the  electromotive  force  in  a 


ELECTROKINETICS  373 

circuit,  whether  it  be  due  to  voltaic  cells,  thermoelectric  junc- 
tions, or  a  dynamo,  in  the  circuit. 

424.  Two  Classes  of  Conductors.  —  Volta  also  sought  to  obtain 
currents  by  circuits  consisting  of  metallic  conductors  only,  but 
in  this  he  did  not  succeed.     He  found,  in  fact,  that,  whatever 
differences  may  be  supposed  to  exist  at  the  various  junctions  in 
such  a  circuit,  the  sum  of  these  formed  as  described  in  §423  is 
equal  to  zero,  that  is,  the  electromotive  force  produced  in  such  a 
circuit  is  zero.     (We  now  know  that  such  is  not  the  case  if  there 
are  differences  of  temperature  in  the  circuit.)     He  was,  therefore, 
led  to  divide  conductors  into  two  classes,  conductors  of  the  first 
class  being  such  as  are  not  competent  by  themselves  to  produce 
an  electromotive  force  when  joined  in  a  circuit,  at  least  one  con- 
ductor of  the  second  class  being  necessary  for  a  finite  electro- 
motive force.     The  former  class  includes  all  metallic  conductors, 
while  the  latter,  now  called  electrolytes  (§462),  are  chemical 
compounds  which  can  be  decomposed  by  an  electric  current. 

It  follows  from  the  above  that  if  we  suppose  the  liquid  of  the 
voltaic  cell  (Fig.  294)  to  be  absent  and  A  and  C  to  be  directly  in 
contact,  then  VDA-h  VAC  +  VCD=0.  We  may  also  evidently 
write  this  equation  in  the  form  VCD  +  VDA  =  VCA,  or  the  contact 
rise  of  potential  from  C  to  A,  if  placed  directly  in  contact,  is 
equal  to  the  sum  of  the  rises  of  potential  from  C  to  D  and  from 
D  to  A,  a  result  that  holds  for  any  three  conductors  of  the  first 
class. 

425.  Electromotive  Force  of  a  Cell.  —  While  an  electromotive 
force  consists  in  general  of  parts  that  are  located  at  different 
points  in  a  circuit  and  is  measured  by  the  work  done  in  taking 
unit  of  electricity  once  around  the  circuit,  it  may,  in  the  case  of  a 
voltaic  cell,  be  considered  as  due  solely  to  the  liquid  and  the 
plates  directly  in  contact  with  it.     For,  from  the  formulas  stated 
in  §§423,  424,  it  follows  that 


—  VAB  +  V  EC  +  VGA 

This  could  not,  however,  be  used  as  the  basis  for  a  satisfactory 
practical  definition  of  the  electromotive  force  of  a  cell,  since,  as 
has  been  stated,  there  is  some  doubt  as  to  the  parts  that  the 
separate  terms  contribute  to  the  whole  sum.  Of  this  whole  sum 


374 


ELECTRICITY  AND  MAGNETISM 


there  is  no  doubt,  since  it  can  be  measured  directly  by  an  elec- 
trometer as  stated  in  §422.  We  shall,  therefore,  define  the  e.m.f . 
of  a  cell  as  follows: 

The  electromotive  force  of  a  voltaic  cell  is  the  difference  of  potential 
of  wires  of  the  same  material  connected  to  the  plates  of  the  cell,  when 
the  cell  is  an  open  circuit. 

When  the  circuit  is  closed  by 
metallic  wires  of  any  kind,  the 
electromotive  force  of  the  circuit 
is,  as  we  have  seen  above,  equal 
to  that  of  the  cell  and  independent 
of  the  material  of  the  connections 
(provided  they  be  all  at  the  same 
temperature) . 

426.  Magnetic  Effect  of  an 
Electric  Current. — In  1820  Hans 
Christian  Oersted,  Professor  in  the 
University  of  Copenhagen,  made 
the  epoch-making  discovery  that 
an  electric  current  acted  on  a  neighboring  magnetic  needle.  It 
was  found  that  when  a  straight  wire  was  held  in  a  north  and 
south  line  over  a  magnetic  needle,  the  needle  was  deflected  if  an 
electric  current  was  passed  through  the  wire.  Further,  the 
direction  of  the  deflection  of  the  needle  was  reversed,  (a)  by 
reversing  the  direction  of  the  current,  and  (b)  by  holding  the 
wire  under,  instead  of  over 
the  needle.  This  is  illustra- 
ted in  Fig.  295. 

The  directions  of  the  cur- 
rents and  of  the  correspond- 
ing deflections  are  described 


s 


Fio.  295. 


N 


FIG.   296. 


by  the  following  rule:  Hold 

the  open  right  hand  on  the 

side  of  the  wire  opposite  the  needle,  with  the  palm  toward  the 

needle,  and  the  fingers  pointed  in  the  direction  of  the  current, 

then  the  thumb  indicates  the  direction  of  the  deflection  of  the 

N  pole  of  the  needle  (Fig.  296). 

Oersted's  experiment  shows  that  a  wire  carrying  an  electric 
current  is  surrounded  by  a  magnetic  field,  and  that  the  direction 


ELECTROKINETICS 


375 


of  the  field  is  on  all  sides  perpendicular  to  the  current  direction, 
that  is,  that  the  magnetic  lines  must  be  circles  about  the  current. 
This  can  also  be  shown  by  means  of  iron  filings.     A  vertical  wire 
passes  through  a  hole  bored 
in  a  horizontal  glass  plate; 
if  a  strong  current  is  passed 

•™^ire" 

'.  fJSZftf&z 

;.',<  *VA^5 
,  i  ;-r  *<i  A  v1 


filings  are  sprinkled  on  the 
glass,  it  is  seen  that  the 
filings  arrange  themselves 
in  circles  with  the  wire  as 
center  (Fig.  297).  By 


using  a 


small 


compass,  it 


is  easy  to  fix  the  direction 
of  this  field.  The  direction 
of  the  current  and  that  of 
the  accompanying  mag- 
netic field  is  stated  by  Max-  FIQ.  297. 
well's  rule:  If  the  direction 

of  the  current  is  that  of  the  advance  or  thrust  of  a  right-handed 
screw,  then  the  direction  of  rotation  of  the  screw  gives  the  direction 
of  the  magnetic  field.  This  is  illustrated  in  Fig.  298. 

From  the  above  we  can  see,  that  a 
N  pole  would  rotate  in  a  circle  about 
a  current,  provided  the  N  pole  could 
be  isolated  from  its  S  pole.  Fig.  299 
shows  a  piece  of  apparatus  for  demon- 
strating this  rotation  and  its  direc- 
tion. The  current  from  the  battery 


Magnetic 


^\roree 

-0- 


Fio.  298. 


Fio.  299. 


enters  from  below  at  A,  passes  up  the  vertical  rod  to  B,  and  by  a  mercury 
cup  enters  the  horizontal  arm  BD;  by  this  it  reaches  the  circular  mercury 
trough  E,  and  completes  the  circuit  back  to  the  battery.  Suspended  BO 


376 


ELECTRICITY  AND  MAGNETISM 


that  they  can  rotate  with  the  arm  BD,  about  the  axis  of  the  vertical  cur- 
rent AB,  are  the  two  magnets  NS  and  N'S',  which  have  their  north  poles 
N  and  N'  in  the  field  of  AB,  while  the  two  south  poles  S  and  S'  are  out- 
side of  this  field.  When  the  current  flows  from  A  to  B,  the  north  poles 
revolve  anti-clockwise  about  the  current  as  looked  at  from  above,  and 
continue  to  revolve  so  long  as  the  current  continues.  Reversing  the 
direction  of  the  current  reverses  the  direction  of  the  rotation. 

427.  Magnetic  Lines  of  a  Circular  Circuit  and  of  a  Solenoid. — 
When  the  wire  carrying  a  current  is  bent  into  a  circle,  as  shown 
in  Figs.  300a  and  &,  the  lines  of  magnetic  force  pass  through  the 


Fio.  300. 


area  bounded  by  the  circle,  entering  at  one  face  of  the  circle  and 
going  out  from  the  opposite  face.  That  is,  the  north  pole  of  a 
magnetic  needle  would  act  as  if  repelled  by  one  face  of  the  circle 
and  attracted  by  the  other  face.  The  circular  circuit  then  acts 
like  a  thin  sheet  magnet,  or  "  a  magnetic  shell,"  one  face  of  which 
is  a  north  "pole"  and  the  other  a  south  "pole."  It  is  also  seen 
that  the  direction  of  the  magnetic  lines  (S  to  N)  of  the  shell  is 
related  to  the  direction  of  the  current  in  the  coil  as  the  thrust  to 
the  twist  of  a  right-handed  screw. 

By  winding  the  wire  closely  on  a  cylinder  in  one  or  more  layers, 
we  get  a  helix  or  solenoidal  coil.     It  can  be  considered  as  a  series 


ELECTROKINETICS 


377 


of  parallel  and  equal  circles  with  centers  on  the  axis  of  the  cylinder. 
A  helix  with  a  current  through  it  forms  a  solenoid.  The  mag- 
netic field  of  a  solenoid  is  indicated  in  Fig.  301.  It  is  the  result- 
ant of  the  magnetic  fields  of  the  individual  circular  currents.  It 
is  seen  that  one  end  forms  a  N  pole,  and 
the  other  end  a  S  pole.  The  magnetic 
field  inside  the  solenoid  is  uniform  ex- 
cept near  the  ends  (§430). 

428.  Current  and  Field  Strength.  Units 
of  Current.  —  The  strength  of  the  mag- 
netic field  at  a  point  P,  due  to  a  current 
t  in  a  small  element  of  circuit  ds  at  a  distance  r  from  P,  (Fig. 
302a)  varies  directly  as  the  current  t,  directly  as  the  length  ds 
resolved  at  right  angles  to  r,  and  inversely  as  the  square  of  the 
distance  r;  that  is,  H,  the  strength  of  the  magnetic  field  at  P,  is 
given  by  the  equation 


Fio.  301. 


where  k  is  a  constant  the  value  of  which  depends  upon  the 
surrounding  medium  and  upon  the  units  used.  The  magnetic 
field  at  P  is  evidently  at  right  angles  to  the  plane  of  Pds.  If 


Fio    302. 


r/* 


'as' 


ds 


ds'  is  the  length  of  the  current  element,  and  if  this  makes  the 
angle  <j>  with  r,  (Fig.  302,  6)  then  ds=dsf  sin  <j>  and 

„     kids'  . 
H  =  —  r—  sin  <z> 
r3 

It  is  evident  that  a  direct  experimental  proof  of  the  above  law  is 
not  possible,  since  we  cannot  have  an  isolated  circuit  element  ds. 


378  ELECTRICITY  AND  MAGNETISM 

We  can,  however,  apply  the  law  to  various  circuits  and  deduce 
formulae  which  can  be  tested. 

The  simplest  application  is  in  calculating  the  magnetic  field  at 
the  center  of  a  circular  circuit.  Here  the  sum  of  all  the  elements 
ds  is  2?rr,  the  circumference  of  the  circle,  and  all  the  elements  are 
at  the  distance  r  from  the  center,  and  hence  the  magnetic  field 
at  the  center  is 

•fj     j  2/m     .2i7tt 

ti.  =Ac  —  =~~n»  — 
ra  r 

In  this  equation,  the  constant,  k  is  unity,  if  we  measure  r  the 
radius  in  centimeters,  and  define  unit  current  as  follows:  Unit 
current  is  the  current  which,  flowing  through  a  circle  of  one  centi- 
meter radius  in  air,  exerts  a  force  of  2n  dynes  on  a  unit  magnetic 
pole  at  the  center  of  the  circle.  This  is  the  c.g.s.  electromagnetic 
unit  of  current.  It  can  also  be  stated  as  follows:  The  electro- 
magnetic unit  of  current  is  that  current  which,  flowing  through 
unit  length  of  arc  with  unit  radius,  produces  unit  magnetic 
field  at  the  center. 

If  the  circular  circuit  has  n  turns  instead  of  one  turn,  the 
formula  becomes, 


By  sending  the  same  current  through  circular  circuits  of  different 
radii  and  measuring  the  magnetic  fields  at  the  centers,  it  is  found 
that  the  above  formula  holds.  It  follows  from  this  that  the 
law  of  action  of  each  short  element  of  a  current  must  also  be  true. 
For  practical  measurements  the  unit  of  current  used  is  the  ampere. 
The  ampere  is  one-tenth  (10"1)  of  the  c.g.s.  electromagnetic  unit 
of  current. 

429.  Electromagnetic  Unit  Quantity  of  Electricity.  —  The  above 
definitions  of  unit  current  are  founded  entirely  on  the  magnetic 
action  of  a  current.  In  stating  them  we  have  implied  nothing 
as  to  the  nature  of  electricity  itself,  the  direction  in  which  it 
flows,  or  the  amount  that  flows.  If,  however,  we  now  assume 
that  an  electric  current  may  be  regarded  as  the  flow  of  an  incom- 
pressible fluid  in  a  definite  channel,  we  must  suppose  that,  as  in 
the  case  of  water  flowing  in  a  pipe,  the  quantity  that  flows 
through  every  cross-section  is  the  same  and  we  are  thus  led  to  an 
entirely  new  definition  of  a  unit  quantity  of  electricity.  Unit 


ELECTROKINETICS  379 

quantity  of  electricity  is  that  quantity  which,  in  each  second,  passes 
through  every  cross-section  of  a  linear  conductor  which  carries  a 
unit  steady  current. 

It  should  be  noted  that  nothing  yet  stated  enables  us  to  decide 
whether  a  current  of  electricity  consists  of  a  flow  of  positive 
electricity  from  high  to  low  potential  in  a  conductor,  or  a  flow  of 
negative  electricity  in  the  opposite  direction,  or  a  combination  of 
both.  This  question  can  only  be  decided  by  considerations  that 
will  be  referred  to  later.  Hence,  the  total  quantity  which 
passes  a  section  of  the  circuit  in 
t  seconds,  when  current  i  flows, 
is  q—it. 

The  c.g.s.  electromagnetic  unit 
of  electricity  is  the  quantity  car- 
ried in  a  second  past  a  point  in 
a  circuit  by  the  c.g.s.  electro- 
magnetic unit  of  current.  Ex- 
periments show  that  this  is  about  FJG  303 

3X1010  times   greater  than  the 

c.g.s.  electrostatic  unit  of  electricity  (§401).  One-tenth  (10"1) 
the  c.g.s.  electromagnetic  unit  quantity  is  called  a  coulomb. 
The  coulomb  is  thus  the  quantity  of  electricity  which  passes 
any  section  of  a  circuit  in  a  second  when  an  ampere  flows  — 
hence  coulombs  =  amperes  X  seconds. 

430.  Magnetic  Field  on  the  Axis  of  a  Circular  Circuit.  Same  for  a  Sole- 
noid. —  The  magnetic  field  at  a  point  P  on  the  axis  of  a  circular  circuit  can 
be  found  as  follows:  Let  p  be  the  distance  of  P  (Fig.  303)  from  C  the 
center  of  the  circle,  and  r  the  radius  of  the  circle.  The  distance  of  P  from 
an  element  ds  of  the  circuit  is  then  a  =  \/r3+pa,  hence  the  magnetic  force 
on  unit  pole  at  P  due  to  current  t  in  ds  is 

„,       ids 
"" 


This  force  is  at  right  angles  to  the  plane  P-ds.     It  is  resolved  along  the  axis 
by  multiplying  by  cos  ^  =  sin  CP4-r/a=-r/\/rs  +  p7;  or 

IT         rids 

"(r'  +  p9)'/1 
For  the  whole  circle  the  intensity  of  the  magnetic  field  is  therefore 

2;rr'i 

"V+p')'/» 
It  is  erident  that  the  component  of  the  force  at  right  angles  to  the  axis  for 


380  ELECTRICITY  AND  MAGNETISM 

each  element  ds  will  be  annulled  by  that  of  the  element  at  the  opposite  end 
of  the  diameter,  and  so  the  above  gives  the  total  field  at  P. 

To  get  the  intensity  of  the  field  at  a  point  P  on  the  axis  of  a  solenoid,  the 
action  of  all  the  parallel  circular  circuits  must  be  added.  Consider  a  small 
section  MM'=*dx  of  the  solenoid,  x  being  the  distance  of  its  center,  C,  from 
P.  Denote  the  number  of  turns  in  the  solenoid  by  n,  its  length  by  L  and  the 
strength  of  the  current  by  t.  The  number  of  turns  in  the  length  dx  is 

MM' 
Aj± 


\C\ 


}OOO<!3OGOGOOOOO  COO  O  O  GO  COO  0  GO  OCX 

Fio.  304.  • 


0' 


ndx/L.     We  thus  get  for  the  intensity  of  the  magnetic  field  at  P  due  to 
this  element  MM'  of  the  solenoid 


Substituting  x  —  r  cot  <j>  and  dx  —  —  rd^/sin*  <f>  the  expression  becomes 

2-irni  sin 


p  _____ 

Integrating  between  the  limits  <£,  —  BPO  and  02=-APO  we  find  for  the 
total  intensity  at  P 

Hp  —  —  !  —  (cos  ^a  —  cos  0J 
I/ 

Now  if  the  length  L  of  the  solenoid  is  large  compared  to  the  radius  r,  and 
we  take  the  point  P  near  the  middle  of  the  solenoid,  we  can  put  fa  —  180° 
and  (/>t=*Q°,  and  hence 


P  --  ZT 

This  formula  also  gives  (approximately)  the  field  intensity  inside  a  ring 
solenoid. 

It  is  to  be  noted  that  at  the  ends  of  a  straight  solenoid,  where  <j)l  or 
<£,=»900,  we  have  #p=27rni/L. 

431.  Field  About  a  Straight  Circuit.  —  The  intensity  of  the  mag- 
netic field  due  to  a  current  in  a  straight  circuit  of  indefinite  length 
varies  inversely  as  the  distance  of  the  point  from  the  circuit. 
This  can  be  proved  experimentally  by  the  arrangement  shown 
in  Fig.  305.  AOB  is  a  vertical  circuit,  and  NS  is  a  magnet 
placed  upon  a  horizontal  disk  which  is  free  to  rotate  about  the 


ELECTROKINETICS 


381 


circuit  as  axis.  If  rt  and  r2  are  the  distances  of  the  poles  +m 
and  —  m  from  the  center  0,  and  Hl  and  #2  the  intensities  of  the 
field  at  the  two  poles,  the  moments  of  the  force  about  0  are 
mHlrl  and  —mH2rv  Experiment  shows  that  there  is  no  rota- 
tion, that  is,  that  the  moments  of  force  are  equal  and  opposite 
or  that  mHlrl  =  mH2r2,  and  hence  Hl/H2  =  rJrr  The  inten- 
sities of  the  field  due  to  the  circuit  are,  therefore,  inversely  as 
the  distances  rl  and  r2.  This  is  known  as  Biot  and  Savart's  law. 
From  this  we  get  that  H  —  ki/r,  for  the  field  about  a  straight 
A 


mH 


FIG.  306 

circuit  of  indefinite  length.  It  can  be  shown  by  mathematics 
and  also  by  experiment,  that  k  =2,  if  i  is  measured  in  c.g.s. 
e.m.  units  of  current,  that  is,  H  =  2i/r. 

The  mathematical  proof  is  as  follows:  The  effect  at  P  (Fig.  306)  of  the 
element  A B  of  the  current  is  the  same  as  that  of  its  projection,  ds  or  AC, 
perpendicular  to  R.  Now  AC**Rd<f>  and  r  =  JB  cos  <f>.  Hence  (see  §428) 
id8/R*=*i  cos  (pd(f)/r.  Integrating  this  between  limits -7T/2  and  +  x/2 
we  get  H>=2i/r. 

We   can   now   calculate  the  work 
done  in  carrying  a  pole  m  round   a 

current  i.    For  W  =  Hml  =  — —   2nr  = 

r 

4mm\  or,  when  a  unit  pole  moves 
about  a  circuit  which  carries  a  cur- 
rent i ,  4m  ergs  work  are  done. 

From  the  above  we  get  an  instructive  proof  of  the  intensity  of 
the  field  at  the  center  of  a  solenoid.  Take  a  closed  path  abed  (Fig. 
307).  The  side  ab  is  a  straight  line  parallel  to  the  lines  of  force. 
The  sides  be  and  ad  are  perpendicular  to  the  field  and  extend 


Fio.  307. 


382  ELECTRICITY  AND  MAGNETISM 

indefinitely,  that  is,  to  a  region  where  the  field  becomes  null. 
Then  in  moving  a  pole  m  around  the  path  abed,  the  only  force  is 
along  the  line  ab.  Hence  the  work  W  =  H.m.ab,  where  H  is  the 
intensity  of  the  field  along  ab.  If  n  is  the  number  of  turns 
in  the  solenoid  and  L  its  length,  the  number  of  turns  in  the  length 
ab  is  n.  ab/L,  and  if  i  is  the  current,  W '  —  knmi.  ab.  n/L.  Hence 

HS=~JJ~'    This    formula    holds    true   approximately   for   any 

point  within  a  long  solenoid  and  not  near  either  end  (§430). 

432.  The  Electric  Current  and  the  Magnetic  Field. — Oersted's 
discovery  shows  that  an  electric  current  is  not  simply  a  transfer 
of  electricity  along  or  in  a  conductor,  but  that  the  whole  region 
about  the  conductor  is  involved.  With  the  transfer  of  electricity 
there  is  a  magnetic  field  at  right  angles  to  the  same.  Professor 
J.  J.  Thomson  has  shown  how  the  various  phenomena  of  the 
electromagnetic  field  may  all  be  interpreted  as  due  to  the  motion 
of  the  electric  lines  or  "Faraday  tubes"  (§418),  which  accom- 
panies the  transfer  of  the  electric  charges.  Suppose  we  have  two 


© 


Faraday  Tube 


Fio.  308o.  FIG.  3086. 

bodies  A  and  B  with  positive  and  negative  charges  respectively. 
These  have  electric  lines  or  tubes  connecting  the  charges.  If  we 
now  connect  A  and  B  by  a  wire  C,  the  ends  of  the  lines  will 
slide  along  the  conducting  wire,  until  the  lines  shrink  to  molecular 
lengths,  when  the  charges  combine.  But  as  these  lines  shrink 
until  their  ends  come  together,  there  are  magnetic  lines  at  right 
angles  to  the  lines  and  to  the  direction  of  their  motion  (Fig.  3086). 
In  the  case  of  a  continuous  electric  current,  there  is  a  continual 
renewal  of  the  electric  lines,  so  that  there  is  a  stream  of  electric 
lines  closing  in  along  the  conductor.  According  to  this  view, 
which  is  in  agreement  with  the  ideas  of  Faraday  and  Maxwell, 
the  magnetic  field  is  due  to  these  moving  electric  lines. 


ELECTROKINETICS  383 

433.  Magnetic  Effect  of  a  Moving  Electrified  Body.— Rowland 
made  in  1875  a  fundamental  experiment  which  showed  that  a 
charged  body  when  moving  at  a  high  speed  is  equivalent  in  its 
magnetic  effects  to  an  electric  current.     His  method  was  to 
charge  a  gilded  vulcanite  disk  and  spin  it  very  rapidly.     This 
produced  a  deflection  of  a  sensitive  magnetic  needle.     Measure- 
ments have  shown  that  a  moving  charged  body  produces  a 
magnetic  field  which  is  equal  to  the  field  produced,  per  unit  of  its 
length,  by  a  linear  conductor  carrying  a  current  eut  where  e  is 
the  charge  and  u  its  speed.     An  electric  discharge  in  a  vacuum 
tube  consists  of  streams  of  charged  particles  called  cathode  and 
canal  rays,  and  it  is  found  that  these  act  in  accordance  with 
Rowland's  experiment,  that  is,  they  are  equivalent  to  electric 
currents,  and  are  bent  and  deflected  by  a  magnet  like  flexible 
currents.     (See  §552  on  Conduction  of  Electricity  in  Gases.) 

434.  Electron  Theory  of  Conduction. — Since  moving  charges  have  the 
same  magnetic  effects  as  electric  currents,  it  is  a  natural  supposition  that 
a  current  consists  essentially  of  a  stream  of  charged  particles,  the  combined 
magnetic    effect  of  which  constitutes  the  magnetic  field  associated  with 
the  current.     In  the  discussion  of  views  as  to  the  nature  of  electric  charges 
(§394)  we  have  seen  that  the  most  probable  hypothesis  is  that  they  consist 
of  electrons  or  units  of  electricity  which  can  be  transferred  from  one  body 
to  another,  an  excess  above  the  normal  constituting  a  negative  charge 
and  a  deficiency  a  positive  charge.     This  hypothesis  has  been  extended  to 
the  explanation  of  electric  currents  and  it  has  been  found  to  account 
fairly  well  for  the  facts.     Since  the  electrons  are  very  much  smaller  than  the 
atoms  of  conductors,  it  would  seem  probable  that  the  current  must  consist 
in  a  flow  of  electrons.     If  so,  the  flow  must  be  from  low  to  high  potentials. 
In  fact,  high  and  low  potentials  have  been  defined  by  the  work  done  in 
moving  a  charge  of  positive  electricity.     If  it  had  been  the  unit  of  negative 
electricity  that  was  referred  to  in  the  definition,  the  terms  high  and  low,  as 
applied  to  the  potential  of  bodies,  would  have  been  reversed. 

It  is  believed  that  in  a  metallic  conductor  many  electrons  are  so  entirely 
"free"  or  so  loosely  connected  to  atoms  that  they  are  easily  set  in  motion 
by  electric  forces,  whereas  the  much  larger  atoms,  each  of  which  remains 
positively  charged  when  deprived  of  its  normal  number  of  electrons,  move 
much  more  slowly.  There  is  evidence  that  the  electrons  are  moving  in 
random  directions  with  very  high  velocities  when  there  is  no  current  in  the 
metal,  for,  under  certain  conditions,  they  can  be  ejected  from  the  surface 
of  the  metal  and  something  can  be  learned  as  to  their  velocities  (§565). 
When  a  difference  of  potential  is  applied  to  the  ends  of  the  conductor,  a 
drift  of  the  electrons  is  superposed  on  their  random  motion  and  this  drift 
constitutes  the  electric  current.  The  drift  or  stream  of  electrons  does  not, 


384  ELECTRICITY  AND  MAGNETISM 

however,  attain  any  very  great  velocity,  since  collisions  between  electrons 
and  atoms  are  continually  taking  place. 

j^From  the  above  we  can  readily  obtain  an  expression  for  the  magnitude 
of  an  electric  current.  The  moving  electrons  in  any  part  of  a  circuit  must 
produce  the  magnetic  field  associated  with  that  part  of  the  circuit.  Now 
consider  a  wire  of  cross-section,  a,  and  let  the  number  of  electrons  per  unit  of 
volume  be  N,  the  charge  of  each  being  e  in  electromagnetic  units.  Then 
the  total  charge  of  these  is  Nae.  Hence,  by  the  result  of  Rowland's 
experiment,  if  the  mean  velocity  in  the  direction  of  the  stream  is  u, 

i  =  Naeu 

This  expression  has  been  tested  in  various  ways,  some  of  which  will  be 
referred  to  later. 

MEASUREMENT  OF  CURRENTS 

436.  Galvanometers. — An  instrument  for  measuring  an  electric 
current  by  its  magnetic  effects  is  called  a  galvanometer.  If  the 
instrument  is  only  for  detecting  the  presence  of  a  current,  it 
should  in  strict  language  be  called  a  galvanoscope,  but  such  an 
instrument  is  often  called  a  galvanometer,  or  perhaps  a  detector 
galvanometer.  There  are  two  types  of  galvanometers  in  com- 
mon use  (a)  the  galvanometer  with  a  movable  magnetic  needle 
and  a  fixed  coil,  and  (6)  the  galvanometer  with  a  movable  coil 
and  a  fixed  magnet.  This  last  type  is  called  the  d'Arsonval 
galvanometer.  Electrodynamometers,  which  are  current  meas- 
uring instruments  depending  on  the  magnetic  action  between  two 
coils,  one  fixed  and  the  other  movable,  form,  strictly  speaking, 
another  type  of  galvanometers  (see  §531). 
The  term  ammeter  or  ampere-meter  is  used 
for  special  forms  of  graduated  galvan- 
ometers. One  of  these  will  be  described 
later  (§440). 

436.  Tangent  Galvanometers. — A  tangent 
galvanometer  consists  of  a  circular  coil, 
which  is  mounted  with  its  plane  vertical 
and  set  in  the  magnetic  meridian,  and  a 
small  magnetic  needle,  suspended  hori- 
FIO.  309.  zontally  at  the  center  of  the  coil.  The 

needle  is  in  a  compass  box  with  a  pointer 
and  graduated  circle  so  that  its  deflections  can  be  read. 
The  deflections  are  often  measured  by  attaching  a  small 
mirror  to  the  needle  and  observing  the  deflection  of  a  beam 
of  light  on  a  scale.  When  an  electric  current  passes  in  the 


ELECTROKINETICS  385 

coil,  the  needle  is  under  the  action  of  the  magnetic  field  of 
the  earth,  which  is  parallel  to  the  coil,  and  of  the  magnetic 
field  due  to  the  current,  which  is  at  right  angles  to  the  coil.  It 
takes  a  resultant  position  and  makes  an  angle  0  with  the  mag- 
netic meridian.  There  are  then  two  couples  acting  on  the  nee- 
dle. The  couple  tending  to  turn  it  back  into  the  magnetic 
meridian  is  HM  sin  6,  where  H  is  the  horizontal  intensity  of 
the  earth's  magnetic  field  and  M  is  the  magnetic  moment  of 
the  needle  (§376).  The  couple  acting  to  turn  the  needle  into 

the  direction  of  the  field  of  the  coil  is  -    -  M  cos  6,  where  - 

is  the  intensity  of  the  field  due  to  the  coil  (§428).  When  the 
needle  is  at  rest,  the  two  couples  are  equal,  or 

,  -  n  Ttir     •       a 

M  cos  0  =  HM  sin  0 


r 


rr 

Hence,  t  -        tan  6 


fi 


2irnr 

r 
FIG.  310. 

In  this  formula,  the  term  -  -  depends  only  on  the  dimensions 

of  the  galvanometer  and  is  represented  by  <?,  called  the  galvan- 
ometer constant.  The  formula  then  becomes 

t  =  #/Gtan  6  =  A  tan  6 

The  current  is  thus  proportional  to  the  tangent  of  the  angle  of 
deflection.  If  the  current  is  to  be  measured  in  amperes,  instead 
of  c.g.s.  electromagnetic  units, 

C  (amperes)  -  -7r-  -  tan  6 


In  the  above  it  has  been  assumed  that  the  magnetic  field  due  to 
the  current  is  uniform  for  the  region  of  the  needle  and  equal  to 


386 


ELECTRICITY  AND  MAGNETISM 


Fia.  311. 


the  magnetic  field  calculated  for  the  center  of  the  coil.     This 
is  approximately  true  when  the  diameter  of  the  coil  is  large 
compared  to  the  length  of  the  needle.     It  is  usual  to  have  a  coil 
25  cm.  or  more  in  diameter,  and  a  needle 
a  centimeter  or  less  in  length. 

In  the  Helmholtz-Gaugain  tangent  galvanometer 
there  are  two  equal  vertical  coils  placed  at  a  dis- 
tance of  the  radius  apart  and  with  the  needle  on 
the  axis  midway  between  the  two  coils  (Fig.  311). 
If  the  coils  have  more  than  a  single  turn  they  are 
generally  wound  on  parts  of  cones  which  have 
their  vertices  at  the  midpoint  of  the  coils.  It  can 
be  shown  that  this  arrangement  gives  a  very 
uniform  magnetic  field  for  the  region  immediately 
around  the  midpoint. 

Tangent  galvanometers  are  used  (1)  to  compare  currents  by 
comparing  the  tangents  of  the  angles  of  deflection  which  they 
produce,  and  (2)  to  measure  electric  currents  in  absolute  units. 
In  the  last  case,  the  values  of  G  and  H  have  to  be  determined. 
To  get  G  is  a  matter  of  simple  measurement  and  arithmetic; 
the  method  of  determining  H  has  been  given  in  §383. 

437.  Sensitive  Galvanometers  (Mov- 
able Needle  Type). — A  tangent  galvan- 
ometer is  primarily  a  standard  in- 
strument. Since  its  coil  must  be  large 
to  give  the  required  uniformity  of  field 
at  the  center,  very  small  currents  will 
not  produce  readable  deflections  of 
the  needle.  To  detect  and  measure 
small  currents  sensitive  galvanometers 
have  been  devised  in  which  (a)  the 
action  of  the  current  on  the  needle  is 
increased  and  (6)  the  directive  action 
of  the  external  field  on  the  needle  is 
weakened. 

Among  the  most  sensitive  of  gal- 
vanometers is  the  astatic  mirror 

galvanometer  of  Professor  William  Thomson,  Lord  Kelvin,  origi- 
nally invented  for  receiving  the  weak  signal  currents  of  the 
Atlantic  cable.  The  magnetic  system  consists  of  two  magnetic 


Fia.  312, 


ELECTROKINETICS 


387 


needles  fixed  parallel  to  each  other  on  the  same  staff,  but  with 
the  poles  of  the  two  needles  oppositely  directed  (Fig.  312).  The 
directive  action  of  the  field  on  the  needle  system  is  thus  propor- 
tional to  the  difference  in  the  magnetic  moments  of  the  two 
needles.  In  this  galvanometer  there  are  two  coils,  one  surround- 
ing the  upper  needle  and  one  surrounding  the  lower  needle  of  the 
astatic  needle  system.  Each  coil  is  double,  and  the  needle  hangs 
between  the  two  parts  of  the  coil.  The  coils  are  usually  wound 
with  very  fine  silk-covered  copper  wire  so  as  to  get  the  maximum 
length  of  wire  on  the  coils  near  the  needles.  This  makes  coils 
of  high  electrical  resistance,  and  hence  such  galvanometers  are 
sometimes  called  "high-resistance"  galvanometers.  The  sug- 
gested term  "long  coil"  galvanometer,  however,  better  describes 


FIG.  31oo. 


FIG.  313&. 


these  instruments.  The  exterior  magnetic  field  is  controlled  in 
direction  and  in  strength  by  one  or  more  controlling  magnets, 
placed  on  top  of  the  instrument.  By  this  means,  the  exterior 
magnetic  field  can  be  reduced  to  any  extent.  A  small  mirror 
is  attached  to  the  needle  system  and  the  deflections  are  read  by 
a  telescope  and  scale,  or  by  a  lamp  and  scale.  For  small  deflec- 
tions the  currents  are  proportional  to  the  deflections.  A  com- 
mon sensitiveness  of  a  galvanometer  of  this  type  is  a  deflection 
of  1  millimeter  of  a  beam  of  light  on  a  scale  at  a  distance  of  1 


388 


ELECTRICITY  AND  MAGNETISM 


meter  for  10~9  amperes  though  galvanometers  of  this  kind  are 
made  with  a  sensitiveness  of  even  10~12  amperes.  Another 
method  of  denoting  the  sensitiveness  of  a  galvanometer  is  to  give 
the  resistance  in  the  circuit  when  an  e.m.f.  of  one  volt  causes  a  de- 
flection of  1  mm.  on  a  scale  at  1  meter  distance.  Thus  a  galvan- 
ometer would  be  described  as  "  sensitive  to  1000  megohms." 

One  of  the  most  sensitive  of  recent  galvanometers  is  the  Broca  galvan- 
ometer. The  needle  system  and  coils  in  this  galvanometer  are  indicated  in 
Fig.  313a.  The  vertical  magnets  have  consequent  poles  and  may  be  very 
strong  and  are  perfectly  astatic  if  parallel.  This  produces  greater  sensitive- 
ness and  greater  freedom  from  external  disturbances. 

In  the  above  sensitive  galvanometers,  some  damping  device  is  necessary, 
in  order  to  bring  the  needle  to  rest.  In  most  cases  air  damping  is  used,  a 
mica  vane  being  for  this  purpose  attached  to  the  needle  system.  Magnetic 
disturbances  due  to  commercial  electric  currents  and  machinery  seriously 
limit  the  use  of  sensitive  astatic  galvanometers;  indeed  in  many  places  they 
cannot  be  used  unless  magnetically  shielded.  This  shielding  is  effected  by 
means  of  a  series  of  hemispheres  or  of  cylinders  of  soft  iron.  (See  §492.) 
Such  instruments  are  called  "iron-clad"  galvanometers. 

The  simple  "astatic  needle  multiplier"  galvanoscope  shown  in  sections  in 
Fig.  3136  was  much  used  by  earlier  investigators.  It  can  be  made  sensitive 
but  is  of  course  affected  by  external  magnetic  fields. 


HO0 


Fio.  314. 


438.  Moving-Coil  or  D'Arsonval  Galvanometers. — A  galvan- 
ometer of  this  type  consists  of  a  small  coil  suspended  be- 
tween the  poles  of  a  strong  magnet  by  a  phosphor-bronze  01 
steel  strip  (Fig.  314).  The  upper  connection  for  the  current  ie 


ELECTROKINETICS 


389 


M 


by  the  suspension  strip,  and  the  lower  connection  is  by  a  loose 
loop  of  fine  copper  wire.  The  "controlling"  force  on  the  coil  is  the 
torsion  of  the  suspension  strip.  When  there  is  no  current  passing 
through  the  galvanometer,  the  plane  of  the  coil  is  parallel  to  the 
magnetic  field.  When  a  current  passes  through  the  coil,  one 
face  becomes  a  north  magnetic  pole  and  the  other  face  a  south 
pole  (§427).  A  current  accordingly  causes  a  deflection  of  the 
coil  in  the  magnetic  field.  In  the  more  sensitive  instruments 
these  deflections  are  read  by  one  of  the  mirror  and  scale  methods. 
The  great  advantage  of  the  d'Arsonval  galvanometer  is  that  it  is 
practically  free  from  external  magnetic  disturbances.  The  quick 
damping  of  the  vibrating  coil  is  also  a 
great  convenience.  This  damping  is 
due  to  the  reaction  of  the  current 
induced  in  the  moving  coil  itself  on 
closed  circuit,  or  in  a  closed  metallic 
frame  inserted  inside  the  coil.  (See 
Lenz's  law  §501.)  .D'Arsonval  gal- 
vanometers are  so  much  more  con- 
venient to  use  than  astatic  galvan- 
ometers, that  they  have  almost  com- 
pletely superseded  the  later  instru- 
ments for  most  electrical  measure- 
ments. A  common  sensitiveness  for 
a  d' Arson val  galvanometer  is  10~8 
amperes  for  1  mm.  deflection  at  a 
meter  distance,  though  in  special  in- 
struments a  sensitiveness  of  10" 10  am- 
peres is  reached. 

In  the  Einthoven  "thread"  galvanometer,  there  is  a  single  fine 
wire  stretched  across  the  field  between  the  poles  of  a  strong  magnet 
(Fig.  315).  A  current  through  this  wire  causes  a  deflection  of 
the  wire,  owing  to  the  mutual  force  between  a  current  and  a 
magnet  (§528).  This  deflection  is  read  by  a  microscope  with 
a  micrometer  eyepiece.  The  magnet  may  be  a  permanent 
magnet  or  an  electromagnet.  Instead  of  a  metallic  wire,  a 
silvered  quartz  fiber  is  used  in  very  sensitive  Einthoven  galvan- 
ometers. A  current  as  small  as  10~12  amperes  can  be  detected 
by  a  galvanometer  of  this  type. 


Fio.  315 


390  ELECTRICITY  AND  MAGNETISM 

439.  Ballistic  Galvanometers. — A  ballistic  galvanometer  is  a 
sensitive  galvanometer  which  has  (1)  a  long  period  of  swing  and 
(2)  little  air  friction  or  other  damping  action  on  the  needle  or 
moving  coil.     Either  type  of  galvanometer  may  be  used  for 
ballistic  purposes.     The  period  suitable  for  most  casep  is  from 
6  to  10  seconds  for  a  single  oscillation.     The  long  period  is  ob- 
tained by  loading  the  needle  or  coil,  thus  increasing  its  moment 
of  inertia.     The  instrument  is  used  to  measure  the  total  quantity 
of  electricity  in  a  transient  current,  such  as  we  get  in  the  dis- 
charge of  a  condenser  or  in  an  induced  current  of  short  duration 
(§505).     The  principle  is  that  the  transient  current  produces 

an  impulse  (§87)  which  is  proportional 
to  it  or  Q;  and  further  that  this  impulse 
is  proportional  to  the  first  throw  6  of  the 
needle,  provided  the  needle  does  not  move 
appreciably  during  the  time  of  the  dis- 
charge. From  this  we  get  Q  —  kdf  or  the 
total  electric  quantity  of  this  discharge 

is  proportional  to  the  first  throw  of  the 

~FIO.  BIG.  needle  or  moving  coil.  The  complete 

theory  of  the  ballistic  galvanometer 

with  discussion  of  the  conditions  of  its  use,  is  given  in  the  larger 

manuals. 

440.  Weston  Ammeter. — This  is  a  D' Arson  val  galvanometer  with  a  coil 
mounted  on  steel  points  in  agate  bearings  and  controlled  by  a  flat  spiral 
spring  (Fig.  316).     The    deflections     \re   read   by   the    movement  of    a 
light   pointer  that   moves   over  a   scale   which  is   graduated  directly  in 
amperes.     The  coil  will  cany  only  very  small  currents,  and  so  shunts 
(§452)  are  used  for  the  larger  currents.     Thus  only  a  known  fraction  of  the 
total  current  passes  through  the  needle  system. 

ELECTROMOTIVE  FORCE  AND  RESISTANCE 

441.  Units  of  Potential  Difference  and  Electromotive  Force. — 
When  a  quantity  of  electricity,  q,  passes  from  a  point  at  the  poten- 
tial 7j  to  a  point  at  the  potential  72,  it  does  an  amount  of  work 
expressed  by  W  =  q(Vl  —  V^  (§402).     When  there  is  a  current 
in  a  conductor  a  quantity  of  electricity,  q  =  it,  flows  in  time  t 
through  every  cross-section  of  the  conductor,  and,  as  regards  the 
part  of  the  conductor  between  points  at  potential  Vl  and  V2, 
the  effect  is  the  same  as  if  the  quantity  q  had  passed  from  one 


ELECTROKINETICS  391 

point  to  the  other.  Hence  the  work  in  this  part  of  the  conductor 
is  W  =  it(V1- 72)  and  therefore  V^V^W/it.  If  W  is  ex- 
pressed in  ergs,  t  in  seconds,  and  i  in  electromagnetic  units  of 
current,  V1  —  V 2  is  in  electromagnetic  units  of  difference  of 
potential. 

The  measure  of  the  electromotive  force  in  a  closed  circuit  has 
been  already  defined  (§423)  as  the  work  done  in  taking  unit 
quantity  of  electricity  once  around  the  circuit.  In  time  t  the 
quantity  q=it  passes  through  every  cross-section  of  the  circuit 
and  the  effect  is  the  same  as  if  q  had  passed  once  around  the  cir- 
cuit. Hence  W  =  Eit  or  E  =  W/it.  Hence  a  difference  of  poten- 
tial and  an  electromotive  force  are  quantities  of  the  same  kind 
and  must  be  expressed  in  the  same  unit.  The  latter  term  is  the 
more  general,  since  it  applies  to  a  complete  circuit,  and  the  electro- 
motive force  in  a  circuit  in  which  the  only  generator  is  a  voltaic 
cell  is  equal  to  the  sum  of  the  potential  differences  at  the 
contacts  (§425).  The  difference  of  potential  between  two  points 
of  a  conductor  which  contains  no  generator  is  frequently  called 
the  electromotive  force  acting  in  that  part  of  the  conductor  or 
simply  the  electromotive  force  between  the  points. 

In  accordance  with  the  equation  E  =  W/it,  stated  above,  we 
define  the  unit  of  electromotive  force  as  follows: 

The  electromagnetic  unit  e.m.f.  exists  between  two  points  when 
one  erg  of  work  is  done  by  one  electromagnetic  unit  of  current  flowing 
for  one  second  between  the  two  points. 

Experiment  shows  that  the  electromagnetic  unit  of  e.m.f.  or 
d.p.  is  about  J  XlO~10  the  electrostatic  unit  d.p.  already  defined 
(§405).  As  a  practical  unit  of  e.m.f.  we  use  the  volt.  The  volt 
is  108  times  the  c.g.s.  e.m.  unit  of  e.m.f.  It  follows  from  the 
above  that  the  product  tiE  is  in  ergs,  when  i  and  E  are  expressed 
in  electromagnetic  units,  and  t  in  seconds.  An  ampere  (10"1  e.m. 
units)  flowing  between  two  points  with  an  e.m.f.  of  one  volt 
(10s  e.m.  units)  will  then  do  107  ergs  of  work  per  second,  or  1 
joule  per  second  (§55).  Thusi  (in  amperes)  X  E  (in  volts)  =  W 
(in  joules  per  second)  =  W  (watts).  This  work  is  transformed 
into  heat  (§458),  or  chemical  energy  (§462)  or  into  mechanical 
energy  (§§534,  536). 

442.  Conductivity  and  Resistance.  Ohm's  Law. — Experiments 
show  that  in  a  circuit  with  a  continuous  and  steady  current,  the 


392  ELECTRICITY  AND  MAGNETISM 

electric  current  is  directly  proportional  to  the  electromotive  force,  the 
constant  of  proportionality  depending  only  upon  the  materials 
and  dimensions  of  the  circuit.  This  fact  is  stated  by  the  equation 
i  =  CE,  where  t  is  the  current,  E  the  e.m.f.,  and  the  constant  of 
proportionality  C  is  called  the  conductance  of  the  circuit.  We 
more  commonly  use  a  different  constant  R,  the  reciprocal  of  the 
conductance  C,  and  write  the  equation  in  the  form  i=E/R.  R 
is  then  defined  as  the  resistance  of  the  circuit.  The  above  is 
called  "Ohm's  law,"  and  was  first  formally  stated  by  G.  S.  Ohm 
in  1828.  If  we  write  the  equation  in  the  form  R  =  E/i,  we  get  the 
important  statement:  the  electrical  resistance  of  a  conducting 
circuit  is  the  constant  ratio  between  the  e.m.f.  and  the  current  in  the 
circuit.  The  resistance  of  a  circuit  therefore  does  not  depend 
upon  the  size  of  the  current;  that  is,  so  long  as  the  dimensions 
and  physical  properties  of  the  circuit  remain  unchanged,  the 
resistance  is  a  constant  for  'the  circuit. 

Ohm's  law  holds  not  only  for  the  whole  circuit,  but  also  for  any 
part.  Thus  if  a  wire  AB,  which  forms  part  of  a  circuit  has  a 
current  *  in  it,  and  an  e.m.f.  or  difference  of  potential  of  EAB 
between  its  ends,  its  electrical  resistance  is  RAB=^AB!^-  From 
this  relation  it  follows  that  R  is  unity  when  E  and  i  are  both 
unity.  That  is,  the  electromagnetic  unit  of  resistance  is  the  re- 
sistance of  a  conductor  in  which  one  electromagnetic  unit  of  current 
is  produced  by  an  electromagnetic  unit  difference  of  potential,  or 
e.m.  unit  e.m.f. 

The  practical  unit  of  resistance  is  the  ohm.  The  ohm  is  the 
resistance  of  a  conductor  in  which  a  current  of  one  ampere  is 
produced  by  a  difference  of  potential  of  one  volt.  This  is  the 
"absolute"  ohm  as  distinguished  from  the  legal  or  international 

ohm  (§449).     We  thus  see  that  R  (ohms)  ast'^<^-     since 

the  volt  =108  c.g.s.  e.m.  units  and  the  ampere  =10~1  c.g.s.  e.m. 
units,  the  ohm  must  equal  109  c.g.s.  e.m.  units  of  resistance. 

443.  Extension  of  Ohm's  Law. — It  is  sometimes  necessary  to  calculate 
the  current  in  a  part  of  a  circuit  when  the  part  in  question  contains  a  voltaic 
cell  (or  some  other  form  of  generator).  For  this  purpose  we  must  use  a 
more  general  form  of  Ohm's  law  which  can  be  found  as  follows.  Consider 
a  circuit  ACBD  which  contains  a  cell,  D,  the  e.m.f.  of  which  is  E.  Applying 
Ohm's  law  to  the  whole  oireuit  and  to  the  part  BCA,  in  which  there  is  no 


ELECTROKINETICS  393 

generator,  and  denoting  the  resistance  of  the  part  ADB  by  RADB  and  that 
of  BCA  by  RBCA*  we  get 

E  -  i(RADB + BBC  A) 

VB-VA-iRBCA 

From  these  we  get  by  subtraction 

E+VA-VB=iRADB 
or  i 

Thus  to  get  the  current  that  flows  from  A  to  B  we  must  add  to  the  excess 
(positive  or  negative)  of  the  potential  of  A  over  B  the  e.m.f.  of  the  generator 
D  and  divide  by  the  total  resistance  of  the 
part  ADB,  including  that  of  the  generator. 
The  above  shows  that,  in  applying  Ohm's 
law  to  a  part  of  a  circuit,  the  electromotive 
force  in  that  part  cannot  be  taken  as  the 
difference  of  potential  at  the  ends  when  the 
part  contains  a  generator.  We  shall  see 
later  that  there  are  cases  in  which  every  part 
of  a  circuit  must  be  regarded  as  a  generator 
(§§499,  500). 

444.  Specific  Resistance.  Conductivity.— The  resistance  of  a 
conductor  varies  directly  as  its  length  L,  inversely  as  its  cross- 
section  A,  and  directly  as  a  quantity  p,  called  the  specific  resist- 
ance or  the  "resistivity"  of  the  material,  that  is  #=^-  The 

specific  resistance  of  a  substance  is  the  resistance  of  a  bar  of  the 
substance  one  centimeter  long  and  of  one  square  centimeter  cross- 
section.  The  table  on  p.  394  gives  the  specific  resistances  at  0°G. 
of  a  number  of  materials  ordinarily  used  in  the  arts.  From  this 
table  and  the  length  and  cross-section  of  a  wire  we  can  calculate 
its  resistance  by  the  above  formula. 

The  specific  resistance  of  a  substance  varies  with  its  tempera- 
ture, and,  in  the  case  of  solids,  with  properties  which  depend 
upon  previous  treatments,  such  as  hardness,  temper,  structure, 
etc.  The  effect  of  temperature  will  be  discussed  in  a  later 
section  (§447).  The  effect  of  previous  treatment  cannot  be 
stated  in  a  simple  form.  Hence  different  samples  of  the  same 
substance  may  show  quite  different  specific  resistances. 

The  specific  electrical  conductivity  of  a  substance  is  the  recip- 


394 


ELECTRICITY  AND  MAGNETISM 


rocal  of  its  specific  resistance.  Thus  taking  the  specific  resistance 
of  certain  copper  as  1.59X10"8,  its  specific  conductivity  is 
l/<o  =6.29X10'. 

SPECIFIC  RESISTANCES 


Material. 

Resistance  in  ohms 
and  cms. 

Temperature 
coefficient  per  C°. 

Aluminum  

3     X10~ 

.0043 

Cooper  .  . 

1  5XlO~ 

.0040 

German  silver  

20     X10~ 

.0004 

Iron  

10.5X10" 

.0062 

Manganin 

42     XlO~~ 

00002 

94     X10~ 

00075 

Platinum 

8  9X10" 

00366 

Silver  

1.5X10" 

.00377 

445.  Weight  of  Wire. — Calculations  of  wire  tables  are  also  conveniently 
made  on  the  basis  of  the  length  and  the  weight  of  the  wire,  the  weight  of 
course  being  directly  proportional  to  the  cross-section  of  the  wire.     The 
reference  unit  in  this  case  is  the  wire  which  is  one  meter  long  and  weighs 
one  gram,  called  the  "meter-gram."     The  Bureau  of  Standards  has  deter- 
mined from  many  experiments  on  commercial  copper,  that  the  "standard" 
meter-gram  of  annealed  copper  wire  at  20°C.  has  a  resistance  of  0.153022 
ohms.     This  corresponds  to  a  density  for  copper  at  20°C.  of  8.89,  and  a 
specific  resistance  of  1.72128  (10)~6  in  ohms  and  centimeters  or  a  specific 
conductivity  of  5.8096  (10)-*  c.g.s.  e.m.  units  at  20°C.     The  mean  tempera- 
ture coefficient  at  20°C.  is  taken  as  «  =  0.00383.     The  conductivity  of  hard- 
drawn  copper  wire  is  about  2.7  per  cent,  less  than  that  of  annealed  copper 
wire.     The  use  of  weights  instead  of  sectional  area  is  to  be  recommended, 
because  the  weight  can  be  determined  with  greater  accuracy,  particularly 
in  the  case  of  fine  wires  and  in  the  case  of  wires  of  irregular  shapes  of  cross- 
sections. 

446.  Resistance  of  Alloys. — The  resistance  of  a  substance  is  in  general 
increased  by  even  small  amounts  of  foreign  substances.     Thus  it  has  been 
found  that  one-half  per  cent,  of  carbon  changes  the  conductivity  of  copper 
by  20  per  cent.     Many  experimenters   have  studied   the   resistances  of 
mixtures  of  metals,  known  as  alloys,  both  on  account  of  the  theoretical 
interest  and  the  practical  importance  of  the  results.     For  one  group  of 
metals,  lead,  tin,  cadmium  and  zinc,  it  is  possible  to  calculate  the  resistance 
of  the  alloy  as  the  mean  resistance  of  the  volume  constituents.     In  the 
case  of  alloys  of  practically  all  other  metals,  the  resistance  of  the  alloy  is 
considerably  greater  than  that  of  any  of  its  constituents.     Another  most 


RESISTANCE  395 

important  property  of  certain  alloys  is  that  the  change  of  resistance  with 
temperature  is  very  small.  These  properties  make  such  alloys  as  constantan, 
manganin,  platinoid,  etc.,  very  valuable  for  resistance  standards  and  for 
rheostats. 

447.  Resistance  and  Temperature.  —  The  electrical  resistance  of 
pure  metals  increases  as  the  temperature  rises.  The  increase 
per  degree  from  0°  to  100°C.  is  a  certain  fraction  of  the  resist- 
ance at  0°C.  This  fraction  is  called  the  temperature  coefficient 
of  resistance.  The  above  law  may  also  be  stated  in  the  form 
of  the  equation,  Rt  =  R0  (1  +  at).  This  when  plotted  gives  a 
straight  line. 

For  larger  ranges  of  temperature  a  formula  involving  a  second 
constant  /?,  and  of  the  form 


must  be  used. 

448.  Resistance  Thermometers.  —  The  resistance  of  a  coil  of 
wire,  being  a  function  of  the  temperature,  can  be  used  to  deter- 
mine temperatures.    Platinum  has  been  found  to  be  the  best 
metal  for  resistance  thermometry.     The  advantages  of  a  plat- 
inum resistance  thermometer  are  sensitiveness,  and  the  wide 
range  of  temperature  that  can  be  measured  (from  the  lowest 
temperature  to  +1200°C.).     A  platinum  thermometer  can  also 
be  of  almost  any  size  and  shape,  and  the  thermometer  coil  can 
be  at  a  distance  from  the  resistance  bridge  and  the  observer. 
Fig.  177  shows  a  standard  resistance  thermometer  as  devised  by 
Callendar. 

449.  Resistance  Standards.  —  It  follows  from  the  definition  of 
the  ohm  that  the  "absolute"  measurement  of  the  resistance  of 
a  conductor  consists  in  determining  the  ratio  of  the  e.m.f.  (volts) 
and  the  corresponding  current  (amperes)  in  the  conductor.     To 
make  such  a  measurement  with  high  accuracy  is  not  a  simple 
process.     But  to  get  the  ratio  of  two  resistances  is,  as  we  shall 
see  later  (§456),  a  relatively  simple  measurement  and  one  that 
can  be  made  easily  with  very  high  accuracy.     Hence  the  ordi- 
nary process  of  determining  the  resistance  of  a  conductor  is  one 
of  comparing  its  resistance  with  a  "standard  resistance." 

Standard  resistances  are  of  two  classes,  (1)  the  prime  stand- 
ard. a  mercury  resistance  and  (2)  secondary  standards  in  the 


306 


ELECTRICITY  AND  MAGNETISM 


FIG.  318. 


I  o 


20 10 

~fo         (Y~ 


Fio.  319. 


form  of  coils  of  wire  either  (a)  single  coils,  or  (6)  groups  of  coils 
mounted  in  boxes  or  cases,  and  hence  called  resistance  boxes. 

^~-^  ^-^        The  prime  standard  is  defined  so  that 

u  n  /Y~u  **  can  e  rePro(iuced  from  the  specifica- 
tions of  materials  and  dimensions  only. 
At  an  International  Congress  of  Electri- 
cians held  at  Chicago  in  1893,  in  which  all 
civilized  nations  were  represented,  it  was 
recommended  that  "the  international  ohm 
be  the  resistance  offered  to  an  unvarying 
electric  current  by  a  column  of  mercury  at 
the  temperature  of  melting  ice,  14.4521  grams  in  mass,  of  a  constant 
cross-sectional  area  and  of  the  length  0/106.3  centimeters."  The 
cross-sectional  area  of  such  a  column 
of  mercury  is  1  square  millimeter. 
This  has  been  adopted  by  all  nations 
as  the  legal  ohm.  The  ohm  as  thus 
defined  by  law  was  as  near  the  ab- 
solute ohm  as  measurements  could 
fix  it  at  the  time. 

Resistances  in  the  form  of  wire  coils  are  the  most  convenient 
working  standards.  First  we  have  single  coils  made  in  a  form 
shown  in  Fig.  318.  They  are  made  so  that  they  can  be  immersed 

in  an  oil  bath  of  constant 
temperature,  and  are  pro- 
vided with  large  copper 
terminals  to  dip  in  mercury 
cups.  Resistances  of  this 
kind  are  used  primarily  for 
calibrating  the  working  re- 
sistance  boxes.  They 
should  be  supplied  with 
certificates  of  calibration 
Fl(3  320  from  one  of  the  national 

calibrating       laboratories, 

such  as  the  U.  S.  Bureau  of  Standards,  The  Reichsanstalt  of 

Germany,  or  the  National  Physical  Laboratory  of  Great  Britain. 

For  general  laboratory  purposes  resistance  coils  are  mounted 

in  boxes  as  shown  in  Fig.  319.     On  the  ebonite  top  there  are  a 


RESISTANCE  397 

series  of  heavy  brass  blocks  and  the  ends  of  the  coils  are  joined 
to  these  blocks,  so  that  the  current  entering  at  one  terminal 
passes  from  block  to  block  through  each  resistance  coil  in  turn. 
Any  coil  can  be  cut  out  of  the  circuit  by  bridging 
the  brass  blocks  with  a  metal  plug  (Fig.  320).  In- 
stead of  plugs,  a  lever  with  sliding  contacts  is  used 
successfully  in  some  recent  resistance  boxes.  Most 
of  the  high-grade  resistance  boxes  are  now  wound 

FIG    321 

with  manganin  wire.  A  resistance  coil  is  always 
wound  inductionless  (Fig.  321),  that  is,  the  coil  is  wound  back  on 
itself  so  as  to  avoid  magnetic  effects  and  self-induction  (§508). 
450.  Resistance  of  Combinations  of  Conductors,  (a)  Series 
Arrangement.  —  The  total  resistance  of  a  number  of  conductors 
connected  "in  series"  is  equal  to  the  sum  of  the  resistances  of 
the  individual  conductors,  that  is,  R=rl+r2  +  •  •  •,  where  12 
is  the  total  resistance,  and  rlt  r2,  r8,  etc.,  are  the  resistances  of 
the  individual  conductors.  For  the  differences  of  potentials  be- 
tween the  ends  of  the  individual  conductors  are  irlt  ir2,  tra,  etc., 
and  the  total  difference  of  potential  is  iR..  Hence  iR=i(rl+rt 
+  •  •  •)  and  JR=r1+r,+ 

(6)  Parallel  Arrangement.  —  When    a    number  of  conductors 

connect  the  same  two  points, 
the  resistance  of  the  combina- 
tion of  conductors  is  given  by 
the  expression  l//2  =  l/r1  +  l/r, 
+  •  •  •  ,  where  R  is  the  re- 
sultant resistance  and  rlf  r,, 
Flo>  322.  rs;  e^c.,  are  the  resistances  of 

the      individual     conductors. 

Let  E  be  the  electromotive  force  between  the  two  points  A,  B 
(Fig.  322)  and  R,  the  resultant  resistance  of  the  separate  resist- 
ances rl9  ra,  r,,  etc.,  in  parallel.  Then  the  currents  in  the  sepa- 
rate branches  are 

*!«=  tf/f*!,  it-B/rn  tt=#/rt,etc. 
or  the  total  current  is 


or 


398 


ELECTRICITY  AND  MAGNETISM 


We  thus  get  the  statement,  the  sum  of  the  reciprocals  of  the  sepa- 
rate resistances  in  parallel  is  equal  to  the  reciprocal  of  the  resultant 
resistance. 

In  the  above  proofs  we  have  assumed,  in  addition  to  Ohm's 
law,  that  the  algebraic  sum  of  the  currents  flowing  toward  a 
point  such  as  A  is  zero,  that  is,  considering  outward-flowing  cur- 
rents as  negative,  +I  —  i1  —  i2  — ia  =  0. 

461.  Kirchhoff's  Laws. — The  laws  for  steady  currents  in  branched  circuits, 
one  of  which  has  been  assumed  above,  have  been  stated  by  Kirchhoff  in 
the  following  general  form:  (1)  The  algebraic  sum  of  the  currents  which 
meet  at  a  point  is  zero,  or  2"i  =  0.  (2)  In  any  closed  circuit,  the  algebraic 
sum  of  the  products  of  the  current  and  resistance  in  each  of  the  conductors 
in  the  circuit  is  equal  to  the  electromotive  force  in  the  circuit,  r1t1-fr2*a  + 
r,t,  -E. 

The  first  law  is  equivalent  to  the  statement  that  when  the  currents 
in  a  network  are  steady  there  is  no  accumulation  of  electricity  at  any 
junction — all  that  flows  in  must  flow  out.  The  second  law  can  be  deduced 


Fio.  323. 


from  the  extended  form  of  Ohm's  law  stated  in  §  443.  For  let  ABC  be 
any  closed  circuit  in  a  complex  network  and  let  the  currents  resistances 
arid  e.m.f.'s  in  the  branches  be  as  indicated  in  Fig.  323.  Then 


It  is  to  be  noted  that,  in  applying  the  second  law,  one  direction  around 
the  circuit  must  be  chosen  as  positive,  and  each  current  and  e.m.f.  must 
be  considered  as  positive  or  negative  according  as  it  is  in  this  or  the  opposite 
direction  respectively. 

452.  Branched  Circuits,  Shunts. — The  principle  of  parallel  circuits  is 
taken  advantage  of  in  shunts  for  apparatus.  Thus  in  the  case  of  a  galvan- 
ometer, it  is  often  necessary  to  measure  a  current  which  is  much  larger 
than  it  is  desirable  to  pass  through  the  instrument.  A  branch  circuit  or 
shunt  of  known  resistance  r,  is  put  in  parallel  with  the  galvanometer  (Fig. 


RESISTANCE  399 

324).  The  current  is  thus  divided  into  t,  and  ig,  where  tc+t'f  — i,  and 
tV»"**V^»  whence  it  readily  follows  that  igU^r*/ (r9+TB).  Thus  to  have 
only  I/ 10  of  the  total  current  pass  through  the  galvanometer,  we  use  a 
shunt  having  1/9  of  the  resistance  of  the  galvanometer. 

453.  Milli-ammeters  as  Voltmeters. — A  milli-ammeter  with  a  high  resist- 
ance in  series  is  used  as  a  voltmeter.     The  instrument  is  joined  in  parallel 
across  the  terminals  of  the  generator  or  circuit  for  which  the  e.m.f.  is  to  be 
determined  (Fig.  325).     The  current 

through  the  milli-ammeter  is  propor- 
tional to  the  e.m.f.  between  its  termi- 
nals, that  is,  i^EJRi.  If  Rv  is  so 
large  that  introducing  it  as  a  branch 
circuit  does  not  change  the  current 
in  the  main  circuit  appreciably,  then  Fio.  325. 

the  readings  of  the  milli-ammeter  are 

practically  proportional  to  e.m.f.  of  the  circuit  for  all  currents,  and  the 
scale  of  the  voltmeter  can  be  graduated  in  volts.  The  resistance  of  the 
Weston  voltmeter  for  150  volts  is  about  15,000  ohms,  and  hence  takes  a 
current  of  0.01  ampere  or  less.  The  change  of  potential  caused  by  intro- 
ducing this  between  the  terminals  of  circuits  of  moderate  resistances  is 
for  most  purposes  negligible. 

454.  Fall  of  Potential  in  a  Circuit. — When  a  current  flows 
through  a  wire,  there  is  a  decrease  or  fall  of  potential  in  the  direc- 
tion of  the  current,  for  otherwise  there  would  be  no  flow  of  electric- 
ity.    Between  any  two  points  x  and  y  of  a  conductor  the  fall  of 
potential   is   E^  —  iR^,   and   hence   we   get  the   statements: 

(a)  With  a  constant  current  the  fall  of  potential  is  proportional 
to  the  resistane  between  the  two  points. 

(6)  With  a  given  resistance  the  fall  of  potential  is  proportional  to 
the  current  between  the  two  points. 

The  above  simple  deductions  from  Ohm's  law  are  used  con- 
tinually in  applied  electricity.  Thus  with  a  given  current  to  be 
transmitted  from  a  machine  to  a  distance,  and  with  a  certain 
allowable  fall  or  "drop"  of  potential,  the  resistance  (and  hence  the 
size)  of  the  conducting  wire  can  be  directly  calculated.  Two 
of  the  most  important  instruments  for  electrical  measurements, 
the  potentiometer  and  the  Wheatstone  bridge,  are  based  directly 
on  the  above  laws.  These  instruments  are  described  in  the  next 
sections. 

455.  The  Potentiometer. — This  instrument  in  its  simplest  form 
consists  of  a  long  uniform  wire  AB  through  which  a  constant 
current  flows  from  a  battery  M  (Fig.  326).     There  is  a  fall  of 


400 


ELECTRICITY  AND  MAGNETISM 


potential  from  A  to  B  and,  the  wire  being  uniform,  the  fall  of 
potential  between  two  points  is  proportional  to  the  length  of  wire 
or  the  resistance  between  the  two  points.  If  two  points  A  and  C 
on  the  wire  be  joined  to  a  galvanometer  G,  there  will  be  a  current 

through  AGO,  as  shown  by 
the  deflection  of  the  galvan- 
ometer. If  we  now  intro- 
duce an  opposing  e.m.f .,  Ex) 
(a  galvanic  cell,  a  thermo- 
element, etc.)  in  the  galvan- 
ometer circuit,  and  find  the 
point  (7,  when  there  is  no 
current  in  the  galvanometer, 
we  know  that  the  fall  of 
potential  between  A  and  C 
is  equal  to  the  e.m.f.  Ex. 
In  the  same  way  we  find  a 
point  D,  such  that  the  differ- 
ence of  potential  between  A  and  D  is  equal  to  the  e.m.f.,  Ey, 
of  a  second  galvanic  cell. 


FIG.  326. 


Hence 


::  resistance  AC  :  resistance  AD? 
::  length  AC  :  length  AD 


In  this  way  two  electromotive  forces  can  be  compared  and  by 
using  a  standard  cell,  such  as  a  Clark  or  a  Weston  cell  (§473) 
of  known  e.m.f.,  we  can  thus  mea- 
sure any  other  e.m.f.     In  poten- 
tiometers of  the  highest  precision, 
the  exposed  wire  is  replaced  by 
resistance  coils  in  a  box. 

466.  The  Wheatstone  Bridge.— 
This  is  an  arrangement  for  getting 
a  proportion  between  four  resist- 
ances. At  a  point  A  (Fig.  327) ,  the 
circuit  divides  into  two  branches, 
ACB  and  ADB.  There  is  the  same  fall  of  potential,  EAB,  along 
each  branch.  Hence  we  can  find  for  any  point  C  on  the  upper 
branch  a  corresponding  point  D  on  the  lower  branch,  such  that 
the  potentials  of  C  and  D  are  the  same.  When  two  such  points 


Fio.  327. 


RESISTANCE 


401 


are  joined  through  a  galvanometer,  the  instrument  shows  no  de- 
flection. Let  rlf  r2,  rs  and  r4  be  the  resistances  of  the  four  parts 
AC,  CD,  AD,  and  DB.  Then  the  current  in  rtand  rais  i'  and 
the  current  in  r8  and  r4  is  i".  The  falls  of  potential  are  \'rlt 
tV,,  i"r  and  t"r4.  Since  C  and  D  are  at  the  same  potential, 


iriY-r'r, 

and  i'r2  =  i"r4 

By  division  we  then  get 

Thus  *  /^y/ 

Hence,  knowing  the  three  resistances  rlt  ra  and  rs,  we  can  get  the 
fourth  resistance,  or,  knowing  the  ratio  rjrl  and  the  resistance 
r3,  we  can  get  the  fourth  resistance  r4. 


<^/u 

,i 

n  

[ 

II  •  •• 

r, 

5  S         5 

C                /* 

5 

u 

V  a  it 
•O 

4       4       * 

b       <J       *       o 

5 

r  5  i         5 

0= 

K)       ft       ft       ft 

F^ 

a       s?       s?  a  

Fio.  328a. 

Figs.  328a  and  3286  show  forms  of  Wheatstone  bridge;  328a 
a  box  bridge,  often  called  a  "post  office"  box  bridge,  3286  the 
"slide  wire"  or  "meter"  bridge.  In  this  last  form,  the  two 
resistances,  rt  and  r,  are  the  two  parts  AC  and  CB  of  the 
uniform  wire  A B.  The  ratio  ra/r4  is  thus  given  by  the  ratio  of 
the  lengths  CB  and  AC. 

It  is  evident  that  the  galvanometer  and  the  battery  can  be  interchanged 
according  to  the  above  explanation  of  the  Wheatstone  bridge.  The  best 
arrangement  for  sensitiveness  depends  upon  the  relative  resistances  of  the 
arms,  galvanometer  and  battery.  The  rule  which  is  proved  in  larger  trea- 
tises, is  that  the  most  sensitive  arrangement,  when  the  galvanometer  re- 
sistance is  greater  than  the  battery  resistance,  is  gotten  by  placing  the 
galvanometer  between  the  junction  of  the  two  higher  resistances  and  the 
junction  of  the  two  lower  resistances. 
M 


402  ELECTRICITY  AND  MAGNETISM 

457.  Electron  Explanation  of  Electric  Resistance. — On  the  electron 
theory  (5  394)  an  electric  current  consists  of  a  stream  of  electrons,  each  of 
which  is  performing  more  or  less  random  motions  but  is,  on  the  whole, 
moving  forward  in  the  direction  of  the  electric  force.  We  must,  however, 
account  for  the  fact  that  the  stream  moves  forward  at  a  steady  rate  and  not, 
as  might  be  expected,  with  an  acceleration.  The  explanation  is  to  be  found 
in  the  frequent  collisions  between  electrons  and  atoms.  Between  two  succes- 
sive collisions  an  electron  has  an  acceleration  in  the  direction  of  the  electric 
force,  but  the  forward  speed  is  being  continually  checked  by  collisions,  as 
when  a  man  seeks  to  make  his  way  rapidly  through  a  crowd.-  Thus  the  for- 
ward motion  is  limited  by  the  average  forward  velocity  attained  between 
collisions,  and  this  is,  of  course,  proportional  to  the  electric  force.  Now  we 
have  already  seen  that  i  is  proportional  to  Neu,  where  u  is  the  average 
forward  velocity  of  the  stream.  Hence,  assuming  that  N  is  constant  in  a 


FIG.  3286. 

conductor  in  a  constant  physical  condition  and  that  e  is  invariable,  we  see 
that  the  current  is  proportional  to  the  electric  force  and  this  is  Ohm's  law. 
In  conductors  of  different  materials  the  frequency  of  collisions  between 
electrons  and  atoms  must  differ  greatly,  depending  on  the  average  distance 
between  atoms.  Hence  under  equal  electric  forces  the  values  of  u  must 
also  differ.  We  have  thus  a  natural  explanation  of  differences  in  conduc- 
tivity and  resistivity.  The  value  of  e  does  not  vary  and  there  is  good 
evidence  that  N  does  not  differ  much  in  good  conductors,  such  as  metals, 
though  it  must  be  very  different  in  very  poor  conductors. 

HEATING  BY  ELECTRIC  CURRENTS 

468.  Joule's  Law. — That  a  current  heats  a  wire  through  which  it 
passes  was  observed  early  and  in  1841  James  Prescott  Joule 
proved  by  experiments  that  the  heat  produced  varied  directly  as 
the  square  of  the  current  and  directly  as  the  resistance  of  the  wire, 
or  H  is  proportional  to  i*R.  It  was  shown  later  that  if  the  heat 
H  is  expressed  in  calories,  R  in  ohms,  and  i  in  amperes,  then 
H  (calories)  =  .238«i^. 

The  above  can  be  deduced  directly  from  the  energy  relations 
involved.  From  the  definitions  of  the  units  of  electromotive 


RESISTANCE  403 

force  and  current  it  follows  that  the  work  in  joules  done  by  a 
current  of  i  amperes,  when  it  flows  for  t  seconds  between  points  the 
difference  of  potential  of  which  is  e  volts,  is  W  =  iet  =  i*Rt  by 
Ohm's  law.  Dividing  this  by  the  mechanical  equivalent  of  heat 
(4.2  joules  per  calorie),  (§290)  we  get  #(cal.)  =W/4.2  =  .238i*Rt. 
Where  electrical  energy  is  transformed  into  heat  in  connecting 
wires,  it  is  ordinarily  a  loss  and  dissipation  of  energy,  and  so  a 
wire  of  as  low  resistance  as  is  economically  profitable  is  used. 
Electric  currents  are,  however,  widely  used  to  produce  heat  for 
important  applications,  such  as  in  electric  lighting  (arc  and  in- 
candescent lamps)  ,  in  furnaces  for  metallurgical  purposes,  cooking, 
etc.,  in  fuses  of  various  kinds  (safety,  blasting,  etc.),  in  hot- wire 
ammeters,  etc.,  etc. 

459.  Incandescent  Lamps. — The  ordinary  incandescent  lamp 
consists  of  a  high-resistance  filament  of  carbon,  tantalum,  or  tung- 
sten, enclosed  in  an  exhausted  glass  bulb,  and  arranged  with 
terminals  so  that  when  the  metallic  base  of  the  bulb  is  inserted 
in  a  socket  connected  to  electric  mains  from  a  power  station,  a 
current    flows    through  the  filament.     The    current  heats  the 
filament   to  incandescence  and  the  filament  thus  becomes   a 
luminous  source.     The  efficiency  of  an  incandescent  lamp,  that 
is,  the  percentage  of  electrical  energy  transformed  into  visible 
luminous  energy,  is  not  high  at  the  best,  but  is  increased  by 
raising  the  temperature  of  the"  filament.     Increased  efficiency 
thus  becomes  largely  a  question  of  finding  filaments  that  will 
stand  high  temperatures.     Tungsten  lamps  require  a  little  over 
a  watt  of  electrical  power  per  candle  power. 

In  the  Nernst  lamp  the  filament  is  a  rod  or  "glower"  made  of  refractory 
earths  (oxides  of  zirconium  and  yttrium)  and  is  a  conductor  only  when 
heated.  The  "glower"  is  heated  by  an  auxiliary  "heater"  until  it  becomes 
a  conductor,  and  it  is  then  maintained  at  incandescence  by  the  current. 
No  exhausted  bulb  is  required  for  this  lamp  since  the  materials  of  the  glower 
do  not  oxidize  in  the  air. 

460.  The  Electric  Arc. — If  two  carbon  rods  AB  and  BC  (Fig- 
329)  are  in  an  electric  circuit,  and  a  current  of  several  amperes 
passes  across  their  contact  point  BC;  it  is  found  that  the  current 
continues  when  the  carbons  are  separated,  leaving  a  gap  of  a  few 
millimeters  between  the  ends  B  and  C.     A  bluish  "arc"  is  formed 
across  the  gap  BC,  and  at  the  same  time  the  ends  of  the  carbons 
become  incandescent.     If  the  current  is  continuous,  the  positive 


404 


ELECTRICITY  AND  MAGNETISM 


Fia.  329. 


carbon  takes  a  cup  form  known  as  a  "crater"  and  is  very  much 
the  hotter  of  the  two  carbons,  and  also  gives  off  more  light.  The 
highest  temperatures  that  have  been  produced  artificially  are 
those  of  the  crater  of  the  electric  arc,  estimated  by  Violle  at 
3500°C.  The  passage  of  the  current  is  through  carbon 
vapor  formed  between  the  carbons. 

The  complete  discussion  of  the  electric  arc  involves 
the  theory  of  the  conduction  of  electricity  through 
gases.  But  two  or  three  facts  may  be  noted.  The 
current  is  carried  by  ions,  or  charged  particles,  due  to 
the  dissociation  of  the  vapors,  and  the  ions  move  faster 
in  one  direction  than  in  the  other.  If  the  anode  (posi- 
tive) is  kept  cold  no  arc  will  be  formed,  but  an  arc  can 
be  formed  with  a  cold  kathode  (negative).  The 
high  temperature  of  the  anode  is  probably  due  to  the 
bombardment  of  the  ions  from  the  kathode.  With  direct  or 
continuous  currents  an  electromotive  force  of  about  45  volts  is 
necessary  to  maintain  the  arc  satisfactorily.  Currents  from  6 
to  50  amperes  are  used,  the  larger  currents  calling  for  larger 
carbons.  Alternating  currents  may  also  be  used,  but  in  the 
case  of  the  alternating  current,  both  electrodes  are  alike,  and 
the  temperature  of  neither  is  as  high  as  that  of  the  anode  with 
the  equivalent  continuous  current. 

The  carbons  are  consumed  by  oxidation  in  the  electric  arc,  the 
positive  carbon  wearing  away  about  twice  as  rapidly  as  'the 
negative  carbon.  The  consumption  of 
the  carbons  is  greatly  decreased  by 
limiting  the  supply  of  air  to  the  arc  as 
in  the  so-called  "enclosed  arc  lamps." 

Arc  lamps  are  supplied  with  a  mechanism 
for  automatically  feeding  the  carbons.  At  the 
start,  this  must  allow  the  carbons  to  come 
together  and  then  must  pull  them  apart — 
called  "striking  the  arc" — and  it  must  also 
hold  the  arc  at  nearly  a  constant  length. 
Fig.  330  shows  the  principle  of  a  device  for  this. 
The  carbon  is  at  one  end  of  a  lever,  and  at  the 

other  end  is  the  movable  iron  core  of  two  solenoids  A  and  B  Excess  cur- 
rent in  the  arc  and  hence  in  the  "series"  coil  A  pulls  the  core  downward 
and  separates  the  carbons.  The  "shunt"  coil  B  acts  oppositely. 


Fia.  330. 


ELECTROLYSIS  405 

The  electric  arc  may  be  formed  between  any  conductors  which  vaporize; 
at  various  times  other  materials  than  carbon,  such  as  magnetite,  have 
been  used  in  commercial  arc  lamps.  In  the  " naming"  arc,  carbons  impreg- 
nated with  certain  salts  are  used  in  order  to  increase  the  luminous  efficiency. 

461.  Kinetic  Theory  of  Heat  Produced  by  an  Electric  Current.— For 
simplicity  let  us  suppose  that  the  electrons  in  a  conductor  are  initially 
wholly  at  rest.  When  an  electric  force  parallel  to  its  length  is  applied  to 
the  conductor,  each  electron  starts  forward,  but  its  forward  motion  is  soon 
checked  by  a  collision  with  an  atom,  and,  since  the  rebounds  of  the  various 
electrons  will  be  in  all  directions,  energy  of  undirected  or  random  motion, 
that  is,  heat,  will  result.  It  is  clear  that  the  same  effect  must  be  continually 
taking  place,  whatever  the  actual  state  of  motion  at  any  moment  may  be. 
On  the  other  hand,  the  random  motion  has  in  itself  no  tendency  to  return 
to  directed  motion.  Thus  heat  is  continually  being  produced  at  the  expense 
of  the  current,  and,  in  the  steady  state  of  the  current,  its  energy  is  con- 
tinually renewed  by  the  agent  (battery,  dynamo,  etc.),  that  keeps  up  the 
electric  field.  It  can  readily  be  shown  that  in  each  second  a  constant 
fraction  of  the  energy  of  the  current  is  changed  into  energy  of  random  motion 
or  heat.  Now  the  kinetic  energy  of  forward  motion  of  the  electrons  is 
proportional  to  u*  and,  as  we  have  seen,  i  is  proportional  to  u.  Hence  the 
heat  produced  per  second  is  proportional  to  i9  and  th  is  Joule's  Law. 


ELECTROLYSIS 

462.  Electric  Conduction  in  Liquids  and  Electrolysis. — Some 
liquids  act  like  metallic  conductors,  that  is,  the  only  change 
produced  in  the  conductor  by  the  passage  of  an  electric  current, 
be  it  either  small  or  large,  is  due  to  the  heat  generated.  Mercury 
and  molten  metals  belong  to  the  above  class.  But  another 
class  of  liquids  show,  not  only  heat  changes,  but  also  chemical 
decomposition,  when  they  are  traversed  by  an  electric  current. 
Substances  which  are  thus  decomposed  by  an  electric  current  are 
called  electrolytes,  and  the  phenomenon  of  chemical  decomposi- 
tion by  an  electric  current  is  called  electrolysis.  Solutions  of 
acids  and  salts,  and  molten  salts  are  electrolytes.  Fig.  331 
represents  a  form  of  electrolytic  cell  which  is  convenient  for 
showing  electrolysis  where  gases  are  to  be  collected.  A  solution 
of  hydrochloric  acid  (HC1  +  H20),  is  contained  in  the  connected 
glass  tubes,  and  the  current  enters  and  leaves  the  solution  by  the 
carbon  terminals  or  electrodes  A  and  K.  The  positive  electrode 
is  called  the  anode,  and  the  negative  the  kathode.  Upon  the 
passage  of  an  electric  current,  hydrogen  escapes  at  the  kathode, 


406 


ELECTRICITY  AND  MAGNETISM 


A 


Cl 


H 


and  is  collected  in  the  glass  tube  above,  and  chlorine  escapes  at 
the  anode.  Since  the  chlorine  gas  is  soluble  in  water,  it  does 
not  appear  in  the  tube  until  the  water  is  saturated.  The  gases 
appear  at  the  electrodes,  and  no  decomposition  appears  in  the 
body  of  the  liquid.  The  part  separated  at  the  anode  is  called 
the  anion,  and  the  part  at  the  kathode  is 
called  the  kation.  The  term  ion  is  used  for 
either  the  anion  or  kation. 

Conduction  in  electrolytes  depends  upon 
the  formation  of  ions.  It  is  assumed  that 
when  hydrochloric  acid,  HC1,  is  put  in  the 
water,  it  is  ionized  or  dissociated,  that  is, 
it  is  broken  into  two  parts  or  ions  which 
have  opposite  electric  charges,  the  hydro- 
gen ion,  H,  which  carries  a  positive  charge 
and  chlorine  ion,  Cl,  which  carries  a  nega- 
tive charge.  Under  the  action  of  the  elec- 
tric forces  from  the  electrodes,  these  mov- 
ing ions  are  directed  into  two  opposite 
streams.  It  is  the  movement  of  these  op- 
posite streams  of  ions  with  their  charges 
that  constitutes  the  electric  current. 
'Electrolytic  conduction  is  thus  a  convec- 
^^  tion  process. 

463.  Secondary  Changes  in  Electrolysis. — In  the  case  described 
above,  the  ions. appear  at  the  electrodes  and  there  is  no  inter- 
mediate chemical  change.  In  many  cases  a  secondary  change 
takes  place.  In  the  electrolysis  of  a  solution  of  sulphuric  acid 
(H2S04  +  H20),  oxygen  is  obtained  at  the  anode,  and  hydrogen 
at  the  kathode,  there  being  two  volumes  of  the  hydrogen  to  one 
volume  of  the  oxygen.  The  accepted  explanation  is  tnat  the 
sulphuric  acid  (H2S04)  is  dissociated  in  the  water  into  ions,  the 
positive  being  H  and  the  negative  S04  or  " sulphion."  Under 
the  directive  action  of  the  charged  electrodes,  a  line  of  the 
hydrogen  ions  is  drawn  to  the  negative  electrode,  where  they  give 
up  their  charges  and  escape  in  bubbles.  Similarly  a  line  of 
sulphion  ions  is  drawn  to  the  positive  electrode,  and  there  they 
give  up  their  charges.  But  the  sulphion  SO4,  cannot  exist  alone, 
and  so  it  replaces  the  O  in  the  water  {H20),  and  0  is  released  in 


FIG.  331. 


ELECTROLYSIS  407 

bubbles  at  the  anode.     The  oxygen  given  off  at  the  anode  is 
thus  the  result  of  a  secondary  chemical  action. 

In  the  case  of  the  passage  of  a  current  through  a  solution  of 
copper  sulphate  CuS04  +  H20,  the  electrodes  being  copper  plates, 
the  CuSO4  is  divided  into  the  ions  Cu  and  S04.  The  Cu  is 
deposited  on  the  kathode,  and  the  SO4,  being  released  at  the 
anode,  combines  with  the  copper  of  the  anode,  forming  GuSO4. 
We  thus  have  the  anode  "wearing  away,"  and  the  "electrolytic" 
copper  deposited  on  the  kathode. 

464.  Dissociation  Theory. — The  theory  of  electrolytic  conduc- 
tion outlined  above  assumes  that  every  molecule  of  the  substance 
before  it  goes  into  solution,  is  made  up  of  two  parts  which  are 
held  together  by  their  opposite  electric  charges,  but  when  it  is 
put  in  water,  the  binding  force  is  decreased  on  account  of  the 
high  dielectric  constant  of  water  (§§401  and  413),  and  so  the  sub- 
stance is  dissociated  into  ions.     These  ions  are  in  constant  motion 
in  all  directions  until  they  are  placed  in  the  electric  field  between 
the  two  electrodes.     By  this  electric  field  they  are  directed  into 
two  opposite  streams,  owing  to  their  electric  charges.     When 
these  streams  of  charged  ions  reach  the  electrodes,  they  lose  their 
charges.     This  decomposition  of  the  electrolyte  in  solution  con- 
tinues as  long  as  the  difference  of  potential  between  the  electrodes 
is  maintained  by  the  external  battery  or  dynamo. 

To  enumerate  the  many  experiments  and  reasons  for  the  above  theory 
of  electrolytic  conduction  is  beyond  the  purpose  of  this  presentation,  but 
two  significant  facts  may  be  mentioned.  By  themselves  the  constituents 
of  an  electrolytic  solution  are  very  poor  conductors.  Thus  water  freed  from 
all  its  impurities  is  a  "non-conductor";  and  pure  sulphuric  acid  is  also  a 
"  non-conductor,"  but  a  solution  of  sulphuric  acid  in  water  is  a  good  con- 
ductor, owing,  as  we  have  seen  above,  to  dissociation  or  ionization.  Again 
Kohlrausch  has  shown  that  the  electrical  conductivity  of  a  dilute  solution 
is  directly  proportional  to  the  number  of  molecules  of  the  salt  or  acid  in 
the  solution,  and  we  are  justified  in  assuming  that  all  the  molecules  are 
ionized,  and  hence  are  carriers  of  electricity.  The  lowering  of  the  freezing 
point  of  solutions,  and  the  phenomena  of  osmosis,  give  added  reasons  for 
the  dissociation  theory  of  solutions,  but  the  discussion  of  these  last  phe- 
nomena belongs  particularly  to  physical  chemistry. 

465.  Ohm's  Law  of  Electrolytes  and  Polarization. — We  have 
seen  that  for  metals,  the  current  is  proportional  to  the  electro- 
motive force,  that  is,  i  =  CE,  where  C  is  a  constant  for  the  circuit. 


408 


ELECTRICITY  AND  MAGNETISM 


When,  however,  an  impressed  e.m.f.,  E,  is  applied  to  the  two 
platinum  electrodes  of  an  electrolytic  cell,  for  example  one  con- 
taining dilute  sulphuric  acid,  it  is  found,  (a)  that  the  current 
starts  at  a  maximum  and  then  rapidly  decreases,  and  (b)  that 
unless  the  e.m.f.  E  is  beyond  a  certain  minimum  value  for  the 
cell,  no  current  is  maintained.  The  current  is  in  fact  propor- 
tional to  (E  —  e)  or  i  =  C(E  —  e).  In  other  words,  there  is  a 
counter  e.m.f.  e,  called  the  electromotive  force  of  polarization. 
The  impressed  e.m.f.  E  must  accordingly  be  greater  than  the 
e.m.f.  of  polarization  if  a  current  is  to  be  maintained.  Thus  for 
a  sulphuric  acid  solution,  there  must  be  an  impressed  e.m.f.  of 
over  1.7  volts  to  maintain  a  current.  The  value  of  e  depends  of 
course,  primarily  upon  the  kinds  of  electrolyte  and  electrodes, 
but  the  size  of  the  current  and  other  conditions  are  factors. 

This  counter  e.m.f.  is  the  result  of  the  formation  of  gas  or  other 
chemical  product  on  the  electrodes.  The  formation  of  the  gas 
layer  has  also  the  effect  of  increasing  the  resistance  of  the  cell. 
The  current  is  thus  decreased  both  on  account  of  the  increas  ed 
resistance  and  on  account  of  the  counter  e.m.f.,  and  both  are 
c  ailed  polarization  effects. 


Fio.  332. 

Kohlrausch  has  shown  that  Ohm's  law  holds  for  electrolytes,  if  the 
polarization  is  eliminated.  This  is  done  by  using  rapidly  alternating 
currents.  In  the  bridge  devised  by  Kohlrausch  for  avoiding  polarization 
and  measuring  the  true  electrolytic  resistance  (Fig.  332),  there  is  the  usual 
slide  wire  bridge  (§456),  but  a  transformer  or  an  induction  coil,  T,  is 
used  in  place  of  a  galvanic  cell.  To  locate  the  point  of  balance  on  the 
bridge,  an  instrument  must  be  used  that  will  detect  small  alternating  cur- 
rents. An  ordinary  galvanometer  will  not  respond  to  alternating  currents 
owing  to  the  inertia  of  the  moving  needle  or  coil,  and  so  a  telephone  is 


ELECTROLYSIS  409 

substituted  for  the  galvanometer.  An  alternating  current  causes  a  buzzing 
in  the  telephone,  and  hence  the  balance  of  the  bridge  can  be  fixed  by  finding 
the  point  on  the  bridge  wire  for  which  there  is  silence  in  the  telephone. 
The  resistance  is  then  calculated  as  usual  from  the  ratios  of  the  bridge  arms. 

466.  Faraday's  Laws  of  Electrolysis. — The  results  of  Faraday's 
experiments  on  electrolysis  are  stated  in  the  following  two  laws : 

I.  The  mass  of  the  substance  liberated  at  an  electrode  is 
proportional  to  the  current  and  the  time,  that  is,  is  proportional 
to  the  quantity  of  electricity  that  passes. 

II.  The  mass  of  the  substance  liberated  by  a  current  per  unit 
time  is  proportional  to  the  chemical  equivalent  weight  of  the 
substance. 

The  first  law  is  expressed  by  the  equation,  W —zit,  where  W 
is  the  number  of  grams  of  the  substance  liberated,  i  the  current 
in  amperes,  and  t  the  time  of  flow  in  seconds.  The  constant  z  is 
the  electrochemical  equivalent  of  the  substance  liberated.  The 
electrochemical  equivalent  of  a  substance  is  therefore  defined  as, 
the  number  of  grams  of  the  substance  liberated  by  an  ampere  in  a 
second,  that  is,  by  a  coulomb.  In  the  last  column  of  the  table 
on  p.  410,  the  electrochemical  equivalents  of  a  number  of  com- 
mon elements  are  given. 

Faraday's  second  law  tells  us  that  if  the  same  current  flows 
through  a  series  of  electrolytic  cells,  these  cells  containing,  for 
instance,  solutions  of  sulphuric  acid,  (H2S04),  silver  nitrate  (Ag- 
N08),  and  copper  sulphate  (CuS04),  there  will  be  liberated  at  the 
anode  8  parts  (by  weight)  of  0,  and  at  the  cathodes,  1.08  parts  of 
H,  107.9  parts  of  Ag,  and  31.8  parts  of  Cu.  These  numbers  are 
proportional  to  the  chemical  combining  quantities  of  the  sub- ' 
stances.  They  can  be  obtained  by  dividing  the  atomic  weights 
by  the  valencies,  as  seen  from  the  table.  Hence,  if  we  know 
the  electrochemical  equivalent  of  one  element,  we  can  calculate 
directly  from  the  atomic  weights  and  valencies  the  electrochemical 
equivalents  of  other  elements.  That  for  silver  has  been  deter- 
mined with  the  greatest  care  at  the  national  physical  labora- 
tories of  the  United  States,  Great  Britain  and  Germany,  and  the 
results  of  the  many  determinations  indicate  the  number  0.0011180 
grams  of  silver  per  coulomb  as  the  electrochemical  equivalent  of 
silver.  To  get  z  for  any  other  element,  we  multiply  this  by 
ratio  of  the  chemical  equivalents.  Where  an  element  has  two 


410 


ELECTRICITY  AND  MAGNETISM 


valencies,  there  will  be  two  values  for  z.  Thus  iron  for  the  ferrous 
salts  has  a  valency  of  2,  and  for  the  ferric  salts  a  valency  of  3, 
with  the  corresponding  values  for  z  as  indicated  in  the  table  below. 


Elements. 

Atomic 
weight. 

Valency. 

Chemical 
equivalent. 

Electro- 
chemical 
equivalent. 

Chlorine. 

35  45 

1 

35  45 

0003672 

Copper.  . 

63.6 

2 

31  8 

000329 

Hydrogen  .  .  . 

1  008 

1 

1  08 

00001044 

Iron,  ferric  

55.9 

2 

27  95 

.  000289 

Iron,  ferrous 

55  9 

3 

18  49 

000193 

Oxygen 

16  0 

2 

8  0 

00008283 

Silver  

107  93 

1 

107  93 

001118 

Zinc 

65  4 

2 

32  7 

000338 

467.  The  Ionic  Charge  or  "Atom  of  Electricity." — If  we  take  the 
same  number  of  grams  of  an  element  as  the  number  denoting 
its  atomic  weight,  and  divide  this  by  the  valency,  we  get  the  gram- 
equivalent  of  the  element.  Thus  the  gram-equivalent  of  silver 
is  (107.93)/!,  and  of  copper  (63.6)/2  or  31.8.  It  is  evident 
from  Faraday's  laws  that  the  quantity  of  electricity  that  depos- 
its the  gram-equivalent  of  one  element  will  deposit  the  gram- 
equivalent  of  every  other  element.  For  silver  this  quantity  is 
107.94^.001118  =  96,550  coulombs.  Hence  96,550  coulombs 
will  deposit  the  gram-equivalent  of  any  element. 

According  to  the  dissociation  theory,  this  charge  is  carried  by 
the  ions.  Hence,  if  we  can  determine  the  number  of  ions  in  a 
gram,  we  can  get  directly  from  the  above  the  electric  charge 
carried  by  a  single  ion. 

By  methods  given  in  special  treatises  on  the  kinetic  theory  of 
gases,  the  number  of  atoms  in  a  gram  of  hydrogen  has  been  cal- 
culated to  be  not  far  from  6X1023.  Hence  the  charge  e  per 
atom  or  ion  for  hydrogen  is  e  =  96,550/n,  or  about  1.6X10"19  cou- 
lombs, or  about  4.8XlO~10  electrostatic  units  of  electricity. 
We  shall  see  later  that  this  same  charge  e  appears  as  the 
unit  charge  in  the  passage  of  electricity  through  gases  (§  564). 
This  is  the  smallest  quantity  of  electricity  that  we  know,  and  all 


ELECTROLYSIS 


411 


Fio.  333. 


other  quantities  appear  to  be  multiples  of  this  unit.  Helmholtz 
as  far  back  as  1881,  noted  the  importance  of  this  in  the  following 
Femarkable  words:  "If  we  accept  the  hypothesis  that  the  ele- 
mentary substances  are  composed  of  atoms,  we  cannot  avoid 
concluding  that  electricity  is  also  divided  into  definite  element- 
ary portions  which  behave  like  atoms  of  electricity."  These 
atoms  of  electricity  we  now  call  "electrons"  (see  §394). 

468.  The    Voltameter    or    Coulombmeter.— The    electrolytic 
cell  gives  us  an  accurate  and  convenient  means  of  measuring 
electric  currents  for  the  calibration  of 

instruments.  An  electrolytic  cell  ar- 
ranged for  measurement  of  currents  is 
called  a  voltameter,  or  perhaps  better 
a  coulombmeter.  The  silver  nitrate 
cell  has  been  found  to  be  capable  of 
such  accuracy  in  measuring  currents 
that  the  international  electrical  con-  -= 
gresses  have  adopted  it  as  a  con- 
venient practical  way  of  defining  the 

"legal"  ampere.  Thus  the  ampere  is  defined  for  practical 
purposes  as  the  current  which  flowing  for  one  second  through 
an  electrolytic  cell  arranged  according  to  directions  fixed  by  law, 
deposits  0.0011180  grams  of  silver  per  second.  Fig.  333  shows  a 
silver  nitrate  coulombmeter. 

PRIMARY  AND  SECONDARY  CELLS 

469.  Simple  Voltaic  Cell. — If  a  plate  of  copper  and  a  plate  of 

zinc  are  dipped  into  dilute  sulphuric  acid  and 
are  connected  by  a  wire  as  shown  in  Fig.  334, 
an  electric  current  is  set  up.  The  current  flows 
through  the  wire  from  the  copper  to  the  zinc, 
and,  in  the  solution  from  the  zinc  to  the  copper. 
The  copper  then  forms  the  positive  pole  but  the 
negative  plate,  and  the  zinc  the  negative  pole 
but  the  positive  plate  of  the  cell.  While  the 
electric  current  flows,  bubbles  of  hydrogen 
appear  on  the  immersed  part  of  the  copper 
plate,  and  the  zinc  plate  wears  away,  zinc  sulphate  being 
formed  and  going  into  solution.  The  above  arrangement  forms 


Cu 


Fio.  334. 


412  ELECTRICITY  AND  MAGNETISM 

the  voltaic  cell  discovered  by  Alexander  Volta  of  Pavia,  Italy, 
in  1800. 

The  first  observations  of  what  we  now  know  as  dynamic  or  sometimes  as 
"galvanic"  electricity,  was  made  by  Galvani,  of  Bologna  in  1789.  He  dis- 
covered that  when  pieces  of  zinc  and  of  copper  were  made  part  of  a  circuit 
with  certain  nerves  and  muscles  of  a  freshly  killed  frog,  that  there  was  a 
contraction  of  the  frog's  muscles.  Galvani  recognized  this  as  electrical,  as 
he  had  produced  similar  effects  with  static  electrical  apparatus.  The 
development  of  Galvani's  discovery  into  the  voltaic  cell  was  made  soon 
afterward  by  Volta. 

The  essential  in  Volta's  cell  is  that  there  are  two  conductors 
And  an  electrolyte  that  acts  chemically  on  one  of  the  conductors 
more  than  on  the  other.  The  number  of  such  voltaic  combina- 
tions that  are  possible  is  indefinitely  large.  It  is  also  possible 
to  have  a  voltaic  cell  by  substituting  a  suitable  electrolyte  for 
one  of  the  metallic  conductors.  While  many  of  these  possible 
cells  are  interesting  and  important  in  the  theory  of  the  voltaic 
cell,  only  a  few  have  practical  value  as  generators  of  electric 
currents. 

In  Volta's  simple  cell,  the  current  from  the  cell  decreases  very 
rapidly  owing  to  the  accumulation  of  hydrogen  on  the  copper 
plate.  The  hydrogen  causes  a  counter  electromotive  force  of 
polarization  and  also  increases  the  internal  resistance  of  the  cell. 
To  reduce  or  eliminate  these  polarization  effects,  and  thus  make 
a  cell  that  will  generate  a  more  or  less  constant  current,  we  have 
two  general  methods,  the  chemical,  and  the  electrochemical 
method.  In  the  chemical  method,  an  oxidizing  agent  is  placed 
round  the  negative  plate,  thus  converting  the  hydrogen  into 
water.  An  example  of  this  is  found  in  the  Leclanche*  cell  de- 
scribed later.  In  the  electrochemical  method  there  are  two 
solutions,  one  around  each  plate,  and  the  hydrogen  combines 
with  the  solvent  around  the  negative  plate  without  freeing  any 
polarizing  products.  The  Daniell  cell,  described  later,  gives  a 
good  example  of  this  method.  The  above  division  of  cells, 
evidently  corresponds  to  a  familiar  division  of  cells  into  "  single 
fluid"  and  "two  fluid"  cells. 

470.  Local  Action. — Commercial  zinc  contains  impurities, 
such  as  particles  of  iron  and  carbon,  and  when  the  zinc  plate  is 
immersed  in  dilute  sulphuric  acid,  these  impurities  form  with 


ELECTROLYSIS 


413 


the  zinc  of  the  plate,  little  local  batteries.  This  "local  action" 
consumes  the  zinc  and  covers  the  plate  with  a  non-conducting 
film  of  gas.  It  has  been  found  that  by  amalgamating  the  zinc 
with  mercury,  local  action  is  largely  eliminated. 

471.  Open  and  Closed  Circuit  Cells.  —  A  cell  which  is  used  only 
at  intervals  and  for  short  periods  has  time  to  recover  from 
polarization  either  by  diffusion  of  the  gas  or  by  the  action  of  an 
oxidizing  material.     Such  a  cell  is  adapted  for  "open  circuit" 
work,  such  as  for  ringing  bells,  etc.,  provided  it  has  the  additional 
property,  that  it  does  not  deteriorate  by  "local  action,"  or  by 
harmful  diffusions.     The  Leclanche*   cell  is  an  example   of  a 
good  open  circuit  cell.      When  more  or  less  current  is  being 
used  continuously,   a   "closed   circuit"   cell   is  needed.     This 
should  have  no  polarization.     The  Daniell  cell,  and  the  lead 
accumulator  or  storage  cell  (§474)  are  examples  of  good  closed 
circuit  cells. 

472.  Two  Typical  Voltaic   Cells.  —  In  this  section  we  shall 
describe  the  Daniell  and  the  Leclanch6  cells,  since  they  are  in  very 
common  use  and  also  typical  cells  for  closed  and  open  circuit  use. 

One  form  of  the  Daniell  cell  is  represented  in 
Fig.  335.  Zn  is  a  rod  of  amalgamated  zinc 
immersed  in  dilute  sulphuric  acid.  This  is  in 
a  porous  cup  C.  Surrounding  the  cup  is  the 
glass  jar  J  which  contains  a  concentrated  solu- 
tion of  copper  sulphate  and  the  copper  plate  Cu. 
The  purpose  of  the  porous  cup  is  to  keep  the 
solutions  from  mixing  and  yet  allow  chemical 
action  between  the  nascent  hydrogen  inside  and 
the  copper  sulphate  of  the  outside  solution. 
When  the  copper  and  zinc  poles  are  connected 
through  an  outside  circuit  R,  an  electric  current  flows  through 
R  from  the  copper  to  the  zinc.  On  the  inside  the  zinc  unites 
with  the  sulphuric  acid,  forming  zinc  sulphate  and  freeing 
hydrogen.  The  hydrogen  replaces  the  copper  in  the  copper  sul- 
phate, and  metallic  copper  is  deposited  on  the  copper  plate.  The 
reactions  are  represented  as  follows: 


FIG.  335. 


Zn  +  H2SO4  =  ZnSO4  +  H2,  inside  the  porous  cup; 
2  +  CuSO4  =  H2SO4  +  Cu  on  the  outside  of  the  porous  cup. 


414 


ELECTRICITY  AND  MAGNETISM 


FIG.  336. 


Zn 


The  gravity  cell,  represented  in  Fig.  336,  differs  only  from  the 
Daniell  cell  in  that  the  separation  of  the  two  solutions  is  main- 
tained by  their  different  densities.  The 
dense  copper  sulphate  occupies  the  bottom  of 
the  jar,  and  the  lighter  acidulated  solution 
rests  above  it.  Under  suitable  conditions 
the  solutions  do  not  mix  to  an  extent  that 
affects  seriously  the  action  of  the  cell. 

The  e.m.f.  of  a  Daniell  or  a  gravity  cell  is 
ordinarily  about  1.08  volts.  Since  there  is  no 
polarization,  the  cell  gives  a  constant  current. 
The  internal  resistance  of  the  cell  is,  how- 
ever, comparatively  large,  so  that  from  a  cell  of  ordinary  size 
only  about  an  ampere  can  be  taken.  The  cell  has  been  largely 
used  in  telegraphy  where  constant  currents 
are  needed. 

The  Ledanche  cell  is  a  single  fluid  cell, 
using  a  solution  of  sal-ammoniac,  and  the 
plates  are  zinc  and  carbon.  The  carbon  is 
enclosed  in  a  porous  cup  and  packed  around 
with  manganese  dioxide  and  broken  carbon. 
The  sal-ammoniac  solution  diffuses  through 
the  porous  cup  to  the  carbon.  The  action  is 
described  as  follows:  The  zinc  unites  with  the 
sal-ammoniac  (NH4C1),  forming  ZnCl4  and  NH3  and  H.  The  hy- 
drogen unites  with  the  Mn02  forming  M203  and  H2O.  The  initial 
e.m.f.  of  this  cell  is  about  1.5  volts.  This  falls 
off  more  or  less  when  the  current  flows,  as  the 
hydrogen  is  not  oxidized  by  the  manganese  dioxide 
as  rapidly  as  formed.  The  cell  recovers,  how- 
ever, when  left  on  open  circuit. 

The  "dry  cell"  which  is  used  so  extensively 
for  spark  coils,  electric  bells,  etc.,  may  be  re- 
garded  as  a  form  of  the  LeclanchS  cell.  The 
zinc  is  in  the  shape  of  a  cylindrical  cup  which 
forms  the  vessel  for  the  cell.  The  carbon  rod 
and  the  oxidizing  dioxide  of  manganese  are  at 
the  center  of  this  cup,  and  are  surrounded  with 
a  packing  of  some  absorbing  substance  such  as  saw  dust.  This 


FIG.  337. 


Zn- 


Fia.  338. 


ELECTROLYSIS  415 

is  saturated  with  sal-ammoniac  solution,  and  the  cup  is  sealed 
with  pitch  to  prevent  evaporation.- 

473.  Standard  Cells  for  E.M.F.  Determinations. — In  calibra- 
tions with  the  potentiometer  (§455)  it  is  necessary  to  have  a 
"normal"  or  "standard"  cell  of  known  and  constant  e.m.f. 
The  two  cells  used  universally  for  this  purpose  are  the  cells 
devised  by  Latimer  Clark  and  by  Edward  Weston.  A  form  of 
the  Clark'cell  is  shown  in  Fig.  339.  The  positive  pole  is  mercury 
(Hg),  in  contact  with  a  paste  of 
mercurous  sulphate  (Hg2S04),  and 
the  negative  pole  is  zinc  in  contact 


Hg2So4 
Paste 


with  the  solution  which  is  zinc  sul- 
phate.    When  this  cell  is  made  strictly    Amza"gam 
according  to  the  specifications  fixed 
by  the  national  physical  laboratories,  FJQ  33Q 

it  has  an  e.m.f.  of  1.434  volts  at  15°C. 
and,  for  a  temperature  t,  an  e.m.f.  of  [1.434-0.0012  (£-15)]  volts. 

The  Weston  cell  is  exactly  like  the  Clark  cell  except  that  the 
zinc  is  replaced  by  cadmium,  and  the  zinc  sulphate  by  cadmium 
sulphate.  Its  e.m.f.  in  the  standard  form  is  1.0190  volts,  and  it 
has  the  great  advantage  of  having  practically  no  change  of 
e.m.f.,  with  temperatures.  No  appreciable  current  should  be 
taken  from  a  standard  cell,  as  the  accompanying  chemical  actions 
cause  more  or  less  permanent  changes  in  the  cell  and  its  e.m.f. 

474.  Storage  Cells. — It  was  noted  that  the  e.m.f.  of  polarization 
in  an  electrolytic  cell  is  due  to  the  gases  or  other  chemical  prod- 
ucts formed  on  the  electrodes.  Thus  in  an  electrolytic  cell 
with  platinum  electrodes  in  dilute  sulphuric  acid,  oxygen  collects 
on  the  anode  and  hydrogen  on  the  cathode,  that  is,  we  get  the 
equivalent  of  a  voltaic  cell,  with  plates  of  oxygen  and  hydrogen. 
When  the  external  current  of  this  electrolytic  cell  is  broken  and 
the  cell  is  joinedHo  a  circuit  containing  a  galvanometer,  we  get  a 
current  from  the  oxygen  "pole"  to  the  hydrogen  "pole,"  that  is, 
opposite  to  the  current  which  produced  the  electrolysis.  A  cell 
thus  formed  by  electrolytic  action  is  called  a  secondary  cell,  in 
distinction  from  voltaic  cells,  which  are  called  "primary"  cells. 
Secondary  cells  are  perhaps  more  commonly  called  storage  cells 
or  electric  accumulators.  It  is  however  to  be  noted  that  the 
energy  "stored"  or  "accumulated"  in  a  storage  cell  is  chemical 
energy  and  not  electrical  energy. 


416  ELECTRICITY  AND  MAGNETISM 

The  current  from  the  "gas"  storage  cell  described  above  is  of 
short  duration,  as  the  gas  layers  are  rapidly  diffused.  In  1860 
Plants  discovered  that,  by  using  lead  plates  in  a  sulphuric  acid 
solution,  a  secondary  battery  could  be  formed  of  large  capacity 
and  one  which  could  be  discharged  days  or  weeks  after  the  time 
of  charge.  In  the  Plante"  cell,  the  surface  of  the  anode  becomes 
coated  with  red  oxide  of  lead,  Pb02,  and  the  hydrogen  escapes 
at  the  cathode.  If,  after  being "  charged,"  the  plates  are 
connected  through  an  external  circuit,  the  chemical  action  of 
charging  is  reversed  and  a  current  is  taken  off  in  the  reverse 
direction  to  the  charging  current.  This  continues  until  the 
products  of  the  electrochemical  decomposition  are  consumed. 
Plants  found  that  the  capacity  of  the  lead  plates  could  be  greatly 
increased  by  a  system  of  charging,  discharging,  and  reversing 
the  charges,  this  "forming"  process  often  taking  many  hours. 
Faure  found  that  he  could  shorten  the  time  of  forming  a  cell  by 
covering  the  anode  with  a  paste  of  peroxide  of  lead.  The  lead 
storage  cell  has  a  normal  e.m.f.  of  about  2  volts,  and  this  e.m.f. 
remains  almost  constant  until  the  cell  nears  the  discharged  con- 
dition. The  internal  resistance  is  low,  and  the  current  output 
is  large. 

For  an  account  of  different  types  of  storage  cells,  and  of  the 
chemical  and  electrical  transformations  involved  in  their  action, 
special  treatises  must  be  consulted. 

476.  Theory  of  the  Voltaic  Cell. — The  decomposition  of  the 
electrolyte  in  the  voltaic  cell  is  the  same  as  that  in  an  electrolytic 
cell;  but,  in  the  case  of  the  electrolytic  cell,  the  e.m.f.  between 
the  electrodes  and  the  energy  of  the  process  is  maintained  by 
an  outside  source,  while  in  the  voltaic  cell  the  e.m.f.  is  produced 
in  the  cell  itself.  The  origin  of  this  e.m.f.  in  the  voltaic  cell  has 
been  one  of  the  debated  problems  of  physics  for. over  a  century. 
There  have  been  two  theories,  the  contact  theory  due  in  its 
original  form  to  Volta,  and  the  chemical  theory  which  was  held 
early  by  Faraday  and  others.  Both  theories  have  naturally 
been  modified  in  various  ways  as  new  facts  have  been  discovered. 
According  to  Volta,  there  is  a  difference  of  potential  between  two 
unlike  bodies,  due  merely  to  their  contact.  By  means  of  a 
sensitive  electroscope  Volta  showed  that,  when  plates  of  zinc 
and  copper  are  brought  into  contact  in  air  and  separated,  the 


ELECTROLYSIS  417 

zinc  becomes  positively  electrified  and  the  copper  negatively 
electrified.  As  the  result  of  his  experiments,  Volta  made  a  series 
in  which  the  metals  are  arranged  in  order  so  that  each  metal  is 
positively  electrified  when  placed  in  contact  with  a  metal  lower 
in  the  series.  Volta's  series  was  zinc,  lead,  tin,  iron,  copper, 
silver,  gold,  carbon.  Later  observers,  using  modern  sensitive 
electrometers,  have  confirmed  the  essential  facts  of  Volta's 
fundamental  experiment.  If  two  metals  given  in  this  series 
form  the  plates  of  a  voltaic  cell,  the  first  in  the  series  forms  the 
positive  plate  and  the  second  the  negative  plate. 

The  interpretation  of  Volta's  contact  experiment  has  been  the 
point  of  controversy.  On  the  chemical  theory,  the  potential  of 
"  contact"  is  due  to  oxidation  by  films  on  the  metals,  the  electrical 
transfer  being  due  to  this  chemical  action.  Since  all  metals 
have  been  in  air,  and  such  invisible  films  are  persistent,  it  has 
been  impossible  to  arrange  test  experiments  free  from  all  ob- 
jection. But  it  is  found  that  metals  boiled  in  a  mineral  oil  at  a 
high  temperature,  which  presumably  removes  any  such  films,  show 
no  difference  of  potential  on  contact,  and  this  is  urged  in  favor  of  a 
chemical  origin  of  the  contact  e.m.f.  It  should  be  noted  that 
even  if  we  hold  that  the  source  of  the  e.m.f.  is  to  be  sought  in  the 
contact  of  unlike  bodies,  the  keeping  up  of  the  electrical  transfer, 
that  is,  of  the  current,  is  due  to  the  chemical  work.  Hence, 
whether  we  think  of  the  chemical  action  as  occasioned  by  the  con- 
tact e.m.f.,  or  think  of  the  e.m.f.  as  due  to  chemical  action,  the 
study  of  the  energy  transformations  is  one  of  chemical  energy. 
The  phenomena  of  these  transformations  in  the  voltaic  cell  are 
of  a  very  complicated  nature.  For  further  discussion  of  this  very 
interesting  subject  the  student  is  referred  to  Nernst's  Theoreti- 
cal Chemistry. 


476.  Series  and  Parallel  Arrangement  of  Cells. — Cells  are  "in 
series"  when  the  positive  pole  of  each  cell  is  joined  to  the  negative 
pole  of  the  next  cell.  The  circuit  is  completed  by  a  conductor  of 

27 


418 


ELECTRICITY  AND  MAGNETISM 


resistance  R  joined  between  the  positive  and  negative  poles  at 
the  ends  of  the  series  (Fig.  340).  The  total  e.m.f.  is  the  sum  of 
the  e.m.f  .'s  of  the  n  cells,  or  is  nE.  The  internal  resistance  is  nr,  so 
that  the  total  resistance  of  the  circuit  is  R  -f  nr.  The  total  current 
is  then,  by  Ohm's  law, 

/--»*- 

R+nr 


Fio.  341. 


Cells  are  "in  parallel,"  when  all  the  positive  poles  are  joined 
together  and  all  the  negative  poles  are  joined  together,  (Fig.  341). 
The  cells  are  thus  equivalent  to  a  single  large  cell  with  an  in- 
ternal resistance  of  r/nt  and  the  e.m.f.  is  E,  that  of  a  single  cell. 
The  current  through  an  external  resistance  R  is  then 


7- 


E 


R+r/n 


Fio.  342. 


Cells  can  also  be  joined  in  a  series-parallel  arrangement  of  p  rows 
in  parallel,  each  row  having  q  cells  in  series  (Fig.  342).  The 
total  number  of  cells  is  n  —  pq.  For  each  row  the  e.m.f.  is  qE,  and 
the  internal  resistance  is  qr.  For  the  p  rows  in  parallel,  the  in- 


THERMOELECTRICITY 


419 


ternal  resistance  is  qr/p,  and  the  e.m.f.  is  qE.     Hence  the  current 
through  R  is 

,         qE  pqE  nE 


R  +  qr/p 


Rp  +  qr 


Iron 


Cold 


The  maximum  current  is  obtained  when  the  cells  are  arranged  so  that  the 
internal  resistance  is  made  as  nearly  equal  to  the  external  resis' ance  as  possible. 

This  is  shown  as  follows:  From  the  equation  I=*(nE)/(Rp  +  qr),  it  is 
evident  that  I  is  a  maximum  when  Rp  +  qr  is  a  minimum.  Write  this  in  the 
form  (\/lftp- \/qr)*  +  2\/Rpqr.  The  term  2\/Rpqr  is  a  constant  for  a 
given  external  resistance  and  a  given  number  of  cells.  Hence  the  value  of 
the  expression  is  least  when  (\/Rp—\/qr)*  =  Q,  that  is,  when/2p=gr,  or 
R  =<r/P-  I*ut  Qr/P  is  the  battery  or  internal  resistance.  Hence  the  current 
is  a  maximum  when  the  internal  resistance  is  equal  to  the  external  resistance. 
Half  of  the  energy  of  the  current  in  this  last  case  goes  into  heating  the  cells. 

THERMOELECTRICITY 

477.  Thermoelectric  Currents. — If  a  copper  and  an  iron  wire 
are  joined  to  form  a  circuit  (Fig.  343)  and  one  junction  of  the  two 
metals  is  heated,  an  electric  current  is 
set  up.  The  current  flows  from  iron 
to  copper  across  the  cold  junction  for 
ordinary  ranges  of  temperature.  Such  Hot 
an  arrangement  of  metals  forms  a 
thermocouple.  This  phenomenon  of 
thermoelectricity  was  discovered  in 
1821  by  Seebeck  who  showed  that 
such  currents  were  produced  by  the  unequal  heating  of  junc- 
tions in  circuits  of  any  dissimilar  metals.  The  electromotive 

forces  produced  in  this  way  are 
very  small,  only  a  small  frac- 
tion of  a  volt  per  couple  in  the 
most  favorable  combinations. 
(See  §479.) 

If  one  junction  of  iron-copper 
couple  is  kept  at  0°  C.,  and  the 
temperature  of  the  other  junc- 
tion   is    raised,    the   e.m.f.    in- 
creases until  a  temperature  of 
This  is  called  the  neutral  tempera- 
e.m.f.  now  decreases  and   becomes 


FIG.  343. 


0    100 


275 
tn 


450  550 
*  HI 


Temp, 


Fio.  344. 


about  275°  G.  is  reached. 
ture  for  the  couple.      The 


420  ELECTRICITY  AND  MAGNETISM 

zero  when  the  temperature  of  the  hot  junction  is  about  550° 
C.  Beyond  550°  the  e.m.f.  is  in  the  inverse  direction,  that  is, 
from  copper  to  iron  across  the  cold  junction.  If  the  tem- 
perature of  the  cold  junction  is  raised,  say  to  100°C.,  the 
temperature  of  inversion  is  lowered  to  450°C.,  that  is,  the  mean 
of  the  temperatures  of  the  cold  and  hot  junctions  at  inversion  is 
equal  to  the  neutral  temperature.  The  general  form  of  the  curve 
for  the  temperature  and  the  e.m.f.  is  shown  in  Fig.  344.  In  this 
tc  and  tw  are  the  temperatures  of  the  cold  and  warm  junctions 
and  tn  the  neutral  temperature.  It  has  been  found  for  most 
cases  to  be  a  parabola,  and  so  the  curve  can  be  determined  for 
a  particular  couple,  if  the  e.m.f.  is  known  for  three  suitable 
temperatures. 

From  the  above  it  is  seen  that  the  thermal  e.m.f.  depends  upon: 

(a)  the  metals  of  the  couple; 

(6)  the  difference  of  temperature  of  the  junctions; 

(c)  the  mean  temperature  of  the  junctions. 

478.  Effect  of  Intermediate  Metals.— In  the  circuit  ABCD  A  (Fig. 
345)  consisting  of  the  two  metals  B  and  D  with  junctions  A  and 
C,  at  temperatures  ^  and  t2  respectively,  let  the  junction  C  be 
broken,  and  let  a  third  metal  X  be  introduced.  If  the  new 


junctions  remain  at  the  temperature  t2,  experiment  shows  that 
the  e.m.f.  of  the  circuit  is  not  changed. 

From  this  we  see  that  the  junction  of  two  metals  for  a  thermo- 
couple can  be  made  either  directly  or  by  solder;  further  that  we 
can  connect  a  galvanometer  in  the  circuit  of  a  thermo-couple  by 
intermediate  wires  without  affecting  the  e.m.f.,  provided  the 
temperature  of  the  junctions  in  the  connecting  circuit  are  kept 
constant.  If  the  temperatures  of  the  new  junctions  are  not 
uniform,  the  effect  is  that  of  introducing  additional  thermo- 
couples into  the  circuit. 


THERMOELECTRICITY 


421 


479.  Thermo-electric  Power  and  the  Thermo-electric  Diagram. — Thermo 
couples  have  found  a  large  use  for  measuring  differences  of  temperature. 
For  this  use  we  want  to  know  the  e.m.f.  per  degree  temperature  difference. 
This  is  called  the  thermo-electric  power  of  the  couple,  and,  as  we  have  seen, 
it  depends  upon  the  mean  temperature  of  the  junctions.  In  Fig.  346,  the 


+  15 


+  10 


250   200 


0    50   700 
Fio.  346. 


200   250 


-20 


-25 


ordinates  represent  the  thermo-electric  powers  of  a  number  of  metals 
referred  to  lead  and  the  abscissas  represent  the  mean  temperatures.  These 
experimental  curves  of  the  thermo-electric  powers  are  practically  straight 
lines  within  these  limits  of  temperature.  To  get  the  thermo-electric  power 
of  any  couple  for  a  given  mean  temperature,  for  example,  an  iron-copper 
couple  for  a  mean  temperature  of  50°,  we  read  the  length  of  the  ordinate 
between  the  iron  and  copper  lines  at  the  abscissa  distance  of  50°.  In  this 


422 


ELECTRICITY  AND  MAGNETISM 


case,  it  is  about  8.7  microvolts  per  degree  difference  in  temperature.  From 
the  points  where  the  lines  of  the  two  metals  intersect,  we  get  the  neutral 
temperature  for  the  couple. 

480.  Peltier  Effect.— Peltier  discovered  in  1834  that  if  a  current  is  sent 
through  the  circuit  of  a  thermo-couple,  heat  is  given  out  at  one  junction 
and  absorbed  at  the  other  junction.     If  the  current  is  reversed,  the  junction 
that  was  heated  is  now  cooled  and  the  other  is  heated.    This  effect  is  due  to 
the  fact  that  at  one  junction  the  current  opposes  the  potential  difference 
between  the  two  metals,  and  hence  work  is  done  there  by  the  current,  and 

this  electric  energy  appears  as  heat.  At  the  other 
junction,  the  potential  difference  of  the  two  metals 
acts  with  the  current  and  there  is  cooling.  To 
show  this  heating  and  cooling  effect  Peltier  made  a 
cross  of  bars  of  antimony  and  bismuth  and  joined 
a  battery  and  a  galvanometer  as  shown  in  Fig.  347. 
When  the  current  flows  from  the  antimony,  DC, 
to  the  bismuth,  CB,  there  is  a  cooling,  as  shown 
by  the  thermo-couple  circuit  CAGE;  when  the 
current  is  reversed,  that  is,  so  as  to  flow  from 

bismuth  to  antimony,  the  galvanometer  shows  a  heating  of  the  junction. 
Tyndall  demonstrated  the  same  phenomenon  by  passing  a  current  through 
an  ordinary  thermo-pile  and  upon  breaking  the  current,  quickly  introduced 
a  galvanometer  in  circuit.  The  galvanometer  showed  an  inverse  current 
in  the  thermo-couple,  corresponding  to  the  unequal  temperatures  at  the 
two  sets  of  junctions  from  the  Peltier  effect  of  the  first  current. 

481.  Thomson  Effect. — Lord  Kelvin  has  shown  that  if  an  electric  current 
is  passed  through  a  bar  along  which  there  is  a  flow  of  heat,  there  is  an  absorp- 
tion or  generation  of  heat  in  the  bar,  which  depends  upon  the  direction  of 
the  current  and  the  nature  of  the  metal.    Thus  if  an  electric  current  passes 
along  a  copper  bar  from  the  cold  to  the  warm  part,  the  copper  is  cooled. 
If  the  current  is  reversed  so  as  to  pass  from  the  warmer  part  to  the  colder, 
the  copper  is  warmed.    In  iron  the  Thomson  effect  is  opposite  to  that  in 
copper.    The  effect  in  lead  is  practically 

zero,  and  hence  lead  is  commonly  used  as 
the  comparison  metal  in  thermo-electric 
diagrams. 

482.  Applications    of   Thermo-couples. — 
As  generators  of  electric  currents,  thermo- 
couples have  little  use  owing  to  their  small 
e.m.f .,  and  their  comparatively  high  internal 
resistance.      But  they  have  found  a  large 

and  valuable  use  for  temperature  determinations,  particularly  for  very 
small  differences  of  temperature,  for  very  high  and  very  low  temperatures, 
and  for  the  temperatures  of  bodies  inaccessible  to  ordinary  thermometers. 
For  small  temperature  differences,  thermo-couples  are  often  arranged  in 
the  form  of  a  thermo-pile.  This  consists  of  alternate  bars  of  the  two  metals 
arranged  in  a  zigzag  order  (Fig.  348)  and  built  into  a  cubical  block  or 


Fio.  348. 


THERMOELECTRICITY 


423 


pile.  The  even-numbered  junctions  form  one  face  of  the  pile,  and  the  odd- 
numbered  junctions  the  opposite  face.  A  difference  of  temperature  between 
the  two  faces  thus  produces  a  series  of  e.m.f.'s,  the  addition  of  which  forms 
the  total  e.m.f. 

One  of  the  most  sensitive  thermo-electric  arrangements  for  detecting 
small  differences  of  temperature  is  Boys'  radio-micrometer.  This  consists 
of  a  single  bismuth  antimony  couple,  the  circuit  of  which  is  completed  by  a 
loop  of  copper  wire,  which  is  suspended  by  a  quartz  fiber  between  the  poles 
of  a  strong  magnet  (Fig.  349).  The  loop  of  wire  thus  forms  the  coil  of  a 
d'Arsonval  galvanometer,  and  gives  a  very  sensitive  means  of  detecting  a 
thermal  current  from  the  antimony-bismuth  couple.  The  couple  is  dia- 
magnetic  and  so  has  to  be  screened  magnetically  by  a  soft  iron  block. 


Fid.  349S 


FIQ.  350. 


With  this  instrument  it  is  said  that  the  heat  of  a  candle  five  hundred  yards 
away  can  be  detected,  or  a  rise  in  temperature  of  less  than  one  millionth  of  a 
degree. 

Duddell  has  made  an  important  application  of  the  Boys'  radio-micro- 
meter in  a  thermo-galvanometer  for  measuring  small  oscillatory  currents, 
such  as  are  used  in  telephony  and  in  the  antennae  of  wireless  telegraphy. 
The  current  heats  a  coil  R  (Fig.  350)  which  is  placed  just  below  the  sus- 
pended thermo-couple  AB.  The  heat  generated  in  R  is  determined  by  the 
deflections  of  the  coil  of  the  radio-micrometer,  and  thus  a  measure  of  the 
current  is  obtained.  Since  the  heating  does  not  depend  on  the  direction 
or  the  frequency  of  the  current,  this  galvanometer  can  be  usedf or  oscillatory 
currents  of  any  period.  Instruments  of  this  kind  have  been  made  sensitive 
to  2.2  XlO-7  amperes. 

For  measuring  temperatures,  from  600°  to  1600°C.,  a  thermo-couple  of 
platinum  and  an  alloy  of  platinum  with  ten  per  cent,  rhodium  is  found  most 


424 


ELECTRICITY  AND  MAGNETISM 


satisfactory.  The  LeChatelier  "thermo-electric  pyrometer"  consists  orf 
such  a  thermo-couple  with  a  suitable  millivoltmeter  graduated  with  a 
temperature  scale.  For  lower  temperatures,  copper-constantan  couples 
are  much  used. 


Fio.  35la. 


ELECTROMAGNETS  AND  MAGNETIC  INDUCTION 

483.  Electromagnets. — A  coil  of  insulated  wire  around  an  iron 
core  forms  an  electromagnet,  when  an  electric  current  flows  in  the 
coil.     The  direction  of  the  magnetization  is  determined  by  the 
direction  of  the  magnetic  field  of  the  coil.     Thus  in  a  helix  A  B 

(Fig.  35 la)  the  current  flows 
anti-clockwise  when  we  look 
at  the  face  B,  and  hence  that 
end  is  a  magnetic  N  face  with 
lines  emerging  as  indicated 
(§427).  The  same  helix  with 
an  iron  core  (Fig.  3516),  shows 
the  same  direction  of  mag- 
netization, since  the  molecular  magnets  of  the  iron  tend  to  line 
up  in  the  direction  of  the  magnetic  field  of  the  coil.  Electro- 
magnets are  used  instead  of  permanent  magnets,  (a)  where  very 
strong  fields  or  very  strong  poles  are  needed;  and  (b)  where  it  is 
desired  to  vary  the  strength  of  the  magnet  or  to  reverse  its 
polarity.  The  latter  can  be  done  by  varying  or  reversing  the 
magnetizing  current.  The  common  uses  of  electromagnets  are 
to  produce  magnetic  fields  as  in  dynamo  machines,  and  to 
exert  forces  as  in  magnets  for 
lifting  loads,  and  in  signaling 
apparatus  (telegraphy,  electric 
bells,  etc.). 

The  design  of  an  electro- 
magnet involves  a  study  of 
the  magnetic  properties  of  the 
iron  core,  and  also  a  calculation  of  the  magnetic  field  of  the  coil. 
The  magnetic  field  of  the  coil  depends  upon  the  dimensions  of  the 
coil  and  the  current,  as  has  already  been  indicated  for  some 
special  cases  (§430). 

484.  Magnetization  of  Iron. — When  a  piece  of  iron  is  placed  in 
a  magnetic  field,  it  becomes  magnetized,  but  only  for  special 


Fia.  3516. 


ELECTROMAGNETS  AND  MAGNETIC  INDUCTION        425 

forms  of  the  specimen  is  the  direction  of  the  magnetization  the 
same  as  that  of  the  external  magnetic  field.  The  induced  poles 
act  against  the  external  field  and  so  have  a  demagnetizing  action, 
in  some  cases,  as  in  that  of  a  short  bar,  this  demagnetizing  action 
is  very  strong.  In  general  the  intensity  and  direction  of  the 
resultant  field,  and  hence  the  magnetization,  does  not  admit  of 
calculation.  In  two  cases  it  is  possible  to  calculate  the  resultant 
magnetizing  force  acting  on  the  iron — (1)  that  of  an  ellipsoid  of 
revolution  with  an  axis  in  the  direc- 
tion of  the  field  and  (2)  that  of  an 
anchor  ring  or  toroid. 

(1)  The   case   of   the   ellipsoid  is 
attained  closely  enough  for  practical 
purposes  by  using  a  cylindrical  rod 
with  its  length  400  or  500  times  its 
diameter,  the  axis  of  the  rod  being 
in  the  direction  of  the  field.     The  uni- 

Fio.  352. 

form  field  is  produced  by  a  solenoid, 

which  is  somewhat  longer  than  the  rod.  The  poles  induced  in 
the  rod  are  so  distant  that  they  do  not  appreciably  change  the 
direction  of  the  field  near  the  middle  of  the  solenoid,  and  there 
the  direction  of  the  field  and  of  the  magnetization  coincide. 

(2)  In  the  second  case,  that  of  the  anchor  ring,  the  magnetic 
field  is  produced  by  an  endless  solenoid  wound  on  the  ring  as 
core  (Fig.  352).     There  are  no  free  poles  in  the  ring  and  hence 
the  field  of  the  solenoid  and  the  magnetization  have  the  same 
directions. 

There  are  two  ways  of  expressing  definitely  the  magnetic  con- 
dition of  iron  or  other  magnetized  substance.  In  the  first  way, 
we  use  a  quantity  called  "  the  intensity  of  magnetization,"  this 
being  represented  by  the  letter  "7."  In  the  second  way,  we 
use  a  quantity  called  "the  magnetic  induction,"  this  being 
represented  by  the  letter  " B."  We  shall  define  "/"  and  " B" 
in  the  following  sections,  and  describe  the  methods  of  testing  the 
magnetic  qualities  of  iron. 

485.  Intensity  of  Magnetization.  IH  Curve  of  Magnetization. — 
Consider  a  small  right  cylinder  of  the  iron  of  volume  v  magnetized 
parallel  to  its  axis.  Let  its  length  be  I  and  the  area  of  its  end  *. 
If  M  is  its  magnetic  moment,  then  M /v,  or  the  magnetic  moment 


426  ELECTRICITY  AND  MAGNETISM 

per  unit  volume,  is  called  the  intensity  of  magnetization,  I,  that  is, 


If  m  is  the  strength  of  the  pole,  M  —  ml  and  v  =  sl.  Hence 
I  =  M/v  =  ml/sl  =  m/s,  where  m  is  the  pole  strength  for  the 
sectional  area  s.  Hence  7,  the  intensity  of  magnetization,  is 
equal  to  the  pole  strength  per  unit  area  of  a  cross-section  at  right 
angles  to  the  magnetization. 

In  the  case  of  a  long  thin  cylinder,  placed  parallel  to  the  field, 
the  magnetization  is  in  the  direction  of  the  field,  and  hence  the 
free  polarity  is  wholly  on  the  ends.  In  this  case,  the  intensity  of 
magnetization  is  I  =  m/nr2,  where  m  is  the  pole  strength  and  ?rra 
is  the  area  of  the  cross-section. 

We  can  measure  the  pole  strength  m  by  the  deflection  of  a 
magnetic  needle  as  described  in  the  next  section.  The  area  rcr3 
is  known,  and  thus  /  can  be  calculated.  The  corresponding 

value  of  H,  the  magnetizing 
force,  is  found  from  the  cur- 

B  /"  rent  and  the  constants  of  the 

solenoid.  The  relation  be- 
tween /  and  H  can  now  be 
shown  by  drawing  a  curve 
with  the  values  of  H  as  ab- 
scissas, and  of  7  as  ordinates. 


1200 
1000 
800 


400 
200 


4  6  8  io  12  i4  16 H  Fig.  353  shows  such  an  IH 
FIO.  353.  curve  of  magnetization  for 

Norway  iron.  At  first,  from 

0  to  A,  the  curve  rises  very  slowly,  and  then  it  rises  rapidly 
to  a  saturation  bend  J5.  The  part  AB  is  almost  a  straight 
line.  From  B  it  rises  very  slowly  for  large  increases  of  the  field 
intensity  H. 

The  explanation  of  this  curve  is  simple.  During  the  first  part, 
that  is  from  0  to  A,  the  groups  of  little  magnets,  formed  by  their 
mutual  attractions,  are  being  broken  up.  As  soon  as  these 
groups  are  broken  up,  the  elementary  magnets  fall  rapidly  into 
the  line  of  the  field,  so  that  at  B,  almost  all  of  them  are  pointing  in 
the  direction  of  the  field. 

The  ratio  of  the  intensity  of  magnetization  to  the  intensity  of 
the  magnetizing  field  is  called  the  magnetic  susceptibility,  k,  of 
the  material.  Hence 


ELECTROMAGNETS  AND  MAGNETIC  INDUCTION        427 


If  the  magnetic  susceptibility  k  were  a  constant,  the  IH  curve  of 
magnetization  would  be   a  straight  line.     It  is  evident  from 
the  experimental  curve  that  k  depends  not  only  on  the  mag- 
netic substance,  but  also  upon 
its  intensity  of  magnetization. 
In  Fig.   354,  we  have  IH 
curves  for  several  materials, 
showing  how  these  substances 
differ  in  their  magnetic  sus- 
ceptibility.    The   results   are 
only    approximate  since   the 
magnetic  properties  of  a  ma- 
terial change  with  treatment. 

486.  Magnetometric   Method  for 
Obtaining  Magnetization  Curves. — 

The  method  as  used  by  Ewing  is  shown  in  Fig.  355a.  The  iron  to  be  tested  is 
a  long  thin  wire,  ns,  in  a  magnetizing  solenoid,  AB.  This  is  placed  vertically 
with  the  upper  end  of  the  wire  ns  on  the  same  level  as  a  small  magneto- 
meter, M,  and  either  east  or  west  (magnetically)  from  M.  The  magneto- 

A 


Fia.  354. 


Fio.  355a. 

meter  consists  of  a  short  magnetic  needle  suspended  by  a  fine  quartz  or 
silk  fiber,  and  supplied  with  a  mirror  for  reading  scale  deflections  with  a 
lamp  or  telescope,  (Fig.  3556).  The  horizontal  field  R  acting  on  M,  due  to 
the  magnetized  wire  ns,  is  calculated  as  follows:  The  intensity  at  M  due 
to  +m  is  m/d2t;  the  intensity  at  M  due  to  —  m,  resolved  horizontally  is 
-m/d8,  (djdj.  Hence  the  total  horizontal  intensity  at  M  due  to  ns  is, 


428  ELECTRICITY  AND  MAGNETISM 

Substituting  for  m  its  value  irr*I,  we  get 


If  H  is  the  intensity  of  the  field  directing  the  magnetometer  "  northward,1 
then  by  the  tangent  law  (§382),  we  have 


or 


i)  H  tan  <f> 


We  can  thus  calculate  I  from  the  deflections  of  M. 


Fia.  3556. 


To  get  the  magnetic  intensity  due  to  ns,  independently  of  the  action  of 
the  solenoid  AB  on  M,  a  compensating  coil  C  carrying  the  same  cur- 
rent as  AB  is  placed  so  that  M  is  not  deflected  by  the  cunents  in  AB 
and  C  when  the  iron  ns  is  not  in  AB.  The  field  H,  is  calculated  from 

the  formula  for  the  solenoid,  H  =  ^~  ( §430) .     The  current  i  is  measured  by 

the  galvanometer  or  ammeter  G,  and  is  increased  or  decreased  by  means  of 
the  rheostat  R.  From  the  values  of  H  and  corresponding  values  of  7  a  curve 
is  plotted. 

487.  Magnetic  Lines  closed  Loops.  Lines  of  Magnetization. 
Magnetic  Induction. — We  have  seen  that  the  space  about  a  mag- 
net is  a  region  of  magnetic  stresses  which  are  shown  by  the  lines 
of  magnetic  force.  As  we  have  described  these  lines,  they 
emerge  at  the  north  pole  and  enter  at  the  south  pole,  their  whole 
course,  so  far  as  it  has  been  described  up  to  this  time,  being  in  the 
air  or  other  non-magnetic  medium.  In  this  respect  the  mag- 


ELECTROMAGNETS  AND  MAGNETIC  INDUCTION        429 

netic  lines  seem,  at  first  sight,  to  be  exactly  similar  to  electro- 
static lines  of  force,  which,  as  we  have  seen,  start  from 
positive  electricity  and  terminate  in  an  equal  quantity  of 
negative  electricity,  and  do  not  continue  into  the  conductors 
(§398).  That  is,  electrostatic  lines  are  not  closed  loops.  We 
shall  show  now  that  the  magnetic  lines  of  a  magnet  are  con- 
tinued into  and  through  the  iron  and  form  closed  loops. 

It  has  been  already  seen  that  the  magnetic  lines  around 
currents  are  closed  loops.  Thus  around  a  linear  circuit  we  have 
circular  lines,  and  about  a  solenoid  lines  which  emerge  from  the 
N  face,  enter  the  S  face  and  complete  the  loop  through  the 
solenoid  (see  Figs.  300  and  301,  §427). 

Consider  now  the  case  of  an  anchor  ring  magnetized  by  a  helix 
or  solenoid  which  is  wound  on  the  ring  as  core.  Here  there  are 
no  poles,  and  so  there  is  no  magnetic  field 
outside  of  the  ring.  But  the  ring  is  mag- 
netized, as  could  be  seen  by  cutting  the  ring 
into  sections.  Each  section,  as  in  the  case  of 
the  broken  magnet  (§366),  would  be  found 
to  be  a  magnet.  Suppose  a  narrow  gap  to  be 
cut  normally  across  the  ring  (Fig.  356).  On 
one  side  of  this  gap  we  find  a  N  pole  and  on 
the  opposite  side  a  S  pole.  If  /  is  the 
intensity  of  magnetization,  and  s  the  area  of  the  section, 
the  pole  strengths  at  the  gap  are  -f  /«,  and  —  Is.  Hence,  4xls 
lines  of  force  (§374)  cross  the  gap  from  the  positive  to  the  nega- 
tive side.  Since  this  will  hold  for  any  and  every  gap  however 
narrow,  we  are  led  to  think  of  these  4nls  lines  as  continuous 
lines  extending  round  the  ring.  In  an  air  gap  these  are  lines  of 
force,  but  in  the  metal,  they  are  called  lines  of  magnetization,  and 
we  may  suppose  them  to  be  due  to  the  lined-up  elementary  mag- 
nets of  the  metal.  But  it  is  to  be  noted  that  a  line  of  force  in  the 
gap  is  the  continuation  of  the  line  of  magnetization  in  the  metal. 

In  addition  to  these  lines  of  magnetization  there  are  lines  of 
force,  H  per  square  centimeter,  due  to  the  magnetizing  solenoid. 
In  the  present  case  the  magnetizing  field  and  the  magnetization 
coincide  in  direction,  that  is,  the  lines  H  and  4nl  (per  square 
centimeter)  are  to  be  added  algebraically.  The  total  number  of 
lines  per  square  centimeter  is  therefore  H  +  4nI.  This  sum  is 


430 


ELECTRICITY  AND  MAGNETISM 


called  the  magnetic  induction,  and  is  represented  by  the  letter 
B.     That  is, 


488.  Magnetic  Flux.—  The  total  number,  N,  of  magnetic  lines 
that  pass  through  an  area,  *S,  normal  to  the  field  is  evidently  equal 

to  the  product  of  the  area  and  the 
magnetic  induction,  that  is, 


B 


A  "tube"  bounded  by  magnetic 
FlQ  357  .    lines   (Fig.   357)   evidently  has  the 

property  that  the  product   of   the 

area  of  any  normal  section  by  the  magnetic  induction,  BS,  is  a 
constant.  This  is  exactly  analogous  to  the  steady  flow  of  a  fluid 
in  a  pipe,  where  the  product  of  the  flow  per  unit  section  and  the 
area  of  the  section  is  a  constant.  Hence,  Maxwell  and  others 
have  used  the  term  magnetic  flux  or  flow  of  magnetic  induction 
for  the  product  BS  =  N.  The  idea  suggested  by  the  term  is  very 
helpful  in  discussing  the  magnetic  induction  in  transformers  and 
other  electromagnetic  apparatus. 

489.  BH  Curves  of  Magnetization. — From  the  IE  curve  of 
magnetization  of  Fig.  353  and  the  relation  B  =  H  +  4nI,  we  can 
calculate   B  and  draw  a  curve 

showing  the  relation  between  B 
and  H  (Fig.  358).  The  BH 
curve  is  in  many  ways  more  val- 
uable than  the  IH  curve,  partic- 
ularly in  connection  with  appa- 
ratus for  electromagnetic  induc- 
tion. In  such  apparatus  we 
need  to  know  BS  —  N,  the  total 
magnetic  lines  or  flux,  and  also 
the  magnetizing  field  H,  which  will  produce  this  magnetic  flux. 

BH  curves  can  be  made  by  the  magnetometric  method  as  de- 
scribed above  (§486),  but  are  as  often  made  by  an  electromag- 
netic induction  method  as  described  later  (§516). 

490.  Magnetic  Permeability. — We  have  seen  that  the  effect  of 
placing  a  piece  of   iron  in  a  magnetic  field  is  to  increase  the 


14000 

12000 

10000 

8000 

6000 

4000 

2000 


468 
Fia.  358. 


10  12  14 


ELECTROMAGNETS  AND  MAGNETIC  INDUCTION 


431 


magnetic  lines  from  H  per  square  centimeter  in  air,  to  B  lines 
per  square  centimeter  in  iron.  Faraday  expressed  this  fact  by 
saying  that  the  iron  has  a  greater  conductivity  for  magnetic  lines 
than  ah*  has.  The  same  fact  is  now  more  frequently  expressed 
by  the  term  magnetic  permeability.  The  magnetic  permeability,  JJL, 
of  a  substance  is  defined  as  the  ratio  of  the  induction  to  the  magnetiz- 
ing force  in  the  substance,  or 

p-B/B 

The  table  below  shows  the  permeability  of  a  specimen  of  iron 
for  various  values  of  H.  It  is  seen  that  beyond  the  bend  of  the 
curve,  where  the  magnetization  approaches  saturation,  the  permea- 
bility decreases  rapidly.  Knowing  the  permeability  of  a  certain 
kind  of  iron  for  a  given  induction,  we  can  find  the  magnetizing 
force  required  to  produce  the  induction.  The  magnetic  suscepti- 
bility k,  and  the  permeability  are  connected  by  the  relation  p  — 
1  +4xk.  This  follows  directly  by  substituting  values  for  B  and  7 
in  the  expression  B  =  H  +  4x1,  and  then  dividing  by  H. 


H 

7 

fc-7 
H 

B 

B 
^H 

0 

0 

0 

0.32 

3 

9 

40 

120 

0.84 

13 

15 

170 

200 

1.37 

33 

24 

420 

310 

2.14 

93 

43 

1,170 

550 

2.67 

295 

110 

3,710 

1,390 

3.24 

581 

179 

7,300 

2,250 

3.89 

793 

204 

9,970 

2,560 

4.50 

926 

206 

11,640 

2,590 

5.17 

1,009 

195 

12,680 

2,450 

6.20 

1,086 

175 

13,640 

2,200 

7.94 

1,155 

145 

14,510 

1,830 

9.79 

,192 

122 

14,980 

1,530 

11.57 

,212 

105 

15,230 

1,320 

15.06 

,238 

82 

15,570 

1,030 

19.76 

,255 

64   . 

15,780 

800 

21.70 

,262 

58 

15,870 

730 

491.  Magnetic  Circuit. — Maxwell's  term  magnetic  flux  or  flow  of  induction 
has  lead  to  the  development  of  the  very  useful  conception  of  a  "magnetic 


432 


ELECTRICITY  AND  MAGNETISM 


circuit"  analogous  to  that  of  an  electric  circuit.  In  an  electric  circuit  w« 
have  an  electric  flux;  that  is,  an  electric  current  i  produced  by  an  electro- 
motive force  E,  in  a  circuit  of  conductance  C  or  of  resistance  R,  where 


R  -  ~  and  the  relation  t  -  CE 


holds  (see  Ohm's  law,  §442).     In  a 


magnetic  circuit  a  magnetic  flux  of  N  is  produced  by  a  magnetizing  field 
(i.e.  "a  magnetomotive  force")  around  a  circuit  of  iron  or  other  materials 
of  greater  or  less  magnetic  permeability.  Let  us  take  the  simple  case  of  an 
iron  anchor  ring  wound  with  a  coil  of  n  turns.  From  §430,  we  have  H  — 


— y— ,  where  L  is  the  length  of  the  mean  length  of  the  core  of  a  circular  sole- 

L 

noid.     Then  the  induction  B  —  pH,  where  ft  is  the  permeability  of  the  core. 
If  S  is  the  area  of  the  cross-section  of  the  core,  we  get  the  total  flux  N  =»  BS 

Substituting  the  value 

This 


can  also  be  written  in  the  form  N 
—  4mt  magnetomotive  force, 
'  L  \  magnetic  resistance 


(S) 


FIG.  359. 


if  we   let    4irm—    m.m.f.    the 

magnetic  motive  force,  and  — •=, 

»S 

equal  the  magnetic  "resistance" 
or  magnetic  reluctance.  The 
magnetic  resistance  of  a  circuit 
thus  varies  directly  as  the  length 
of  the  circuit,  inversely  as  the 
cross-section,  and  as  some  con- 
stant f— J  which  we  may  call  the 

specific  magnetic  resistance  of 
the  material. 

This  parallelism  to  the  electric  circuit  can  be  extended  to  a  composite 
circuit,  where  there  are  two  air  gaps  as  in  the  case  of  the  magnetic  field  of 
a  dynamo,  and  in  such  cases  becomes  of  great  convenience  in  calculations. 

492.  Effects  of  High  Permeability.  Magnetic  Shielding. — When 
a  magnetic  substance  is  placed  in  a  magnetic  field,  it  changes  the 
distribution  of  lines  of  force.  This  is  shown  by  the  arrangement 
of  iron  filings  about  an  iron  disk  placed  in  a  uniform  field 
(Fig.  359).  The  lines  tend  to  go  through  the  iron,  because  of  its 
higher  permeability.  Fig.  360  taken  from  a  paper  of  Lord 
Kelvin,  shows  the  field  around  and  through  a  sphere  of  high 
permeability. 

Fig.  234e  (§368)  shows  the  lines  about  a  thick  iron  ring  in  a 


ELECTROMAGNETS  AND  MAGNETIC  INDUCTION        433 

uniform  magnetic  field.  It  is  to  be  noticed  that  the  lines  are 
diverted  toward  the  ring  on  account  of  its  higher  permeability; 
and  also  that  the  filings  on  the  inside  of  the  ring  show  little  if 
any  directive  action.  This  "screening"  effect,  of  soft  iron  is 
made  use  of  in  protecting  sensitive  galvanometers  from  magnetic 
disturbances  (§437).  The  same  principle  is  used  in  shielding 
the  interior  coils  of  the  ring  armature  (§519) . 

493.  Magnetization    and    Temperature. — When    soft    iron    is 
raised  to  a  temperature  of  785°C.,   it  ceases  to  be  ferro-mag- 
netic,  though  remaining  slightly  magnetic.     This  temperature  is 
called   the    critical    temperature.      For 

nickel,  the  critical  temperature  is  340° 

C.,  for  cobalt,    1075°C.,    for  magnetite 

535°C.,    and    for    a    certain    hard  steel 

Hopkinson    found    a    value    of    690°C. 

The   permeability   of   a   substance   for 

strong  magnetizing  forces  in  general  de-  FlQ  360 

creases  as  the  temperature  rises,  though 

there  are  some  variations  from  this  that  are  not  yet  explained. 

494.  Diamagnetic  Substances. — A  bar  of  bismuth,  suspended 
between  the  pointed  poles  of  a  strong  electromagnet  takes  a 
position  at  right  angles  to  the  lines  of  force,  and  a  small  ball  of 
bismuth  is  repelled  from  strong  to  weak  parts  of  the  field.     This 
action  is  just  opposite  to  that  of  iron  in  a  magnetic  field,  and 
hence  Faraday  called  bismuth  and  other  bodies  showing  similar 
magnetic  action,  diamagnetic  substances.     Bismuth,  antimony, 
copper,  zinc,  silver,  lead,  glass,  etc.,  are  diamagnetic.    The  action 
of  bismuth,  the  strongest  of  the  diamagnetic  substances  is  how- 
ever feeble  compared  to  the  magnetic  action  of  iron,  nickel  and 
cobalt. 

495.  Para -magnetic  and  Ferro -magnetic  Substances. — Faraday 
found  that  many  bodies,  supposed  to  be  non-magnetic  show 
magnetic   properties   in   the  field   of   a  strong  electromagnet. 
Such  bodies  are  called  para-magnetic  substances.     The  strongly 
para-magnetic  substances,  iron,  nickel,  cobalt,  Heusler  alloy,  are 
known  as  ferro-magnetic  substances.     The  following  table  gives 
the  susceptibilities  of  some  substances,  negative  susceptibilities 
indicating  diamagnetic  substances.     The  susceptibility  of  the 
ferro-magnetic  substances  as  WB  have  seen  depends  upon  the 


434  ELECTRICITY  AND  MAGNETISM 

magnetizing  force.  As  the  susceptibility  is  changed  by  slight 
impurities,  the  values  given  by  different  observers  vary  some- 
what. Since  the  permeability  /z  is  equal  to  1+47T&,  it  is  seen 
that  negative  susceptibilities  indicate  permeabilities  less  than 
unity. 

Iron-silicon  alloy  melted  Aluminum 0.0000018 

in  a  vacuum 40,000+           Air 0.000000032 

Ordinary  soft  iron,  max.  200+                 Copper -0.00000082 

Nickel,  max 23+                 Lead -0.00000124 

Cobalt,  max 13.8+             Silver -0.00000151 

Oxygen  at  182°  C 0 . 000324       Antimony -  0 . 0000052 

Platinum 0.000029       Bismuth -0.0000138 

496.  Ampere's  Theory  of  Magnetism.  Electron  Theory  of 
Magnetism. — The  "molecular"  theory  of  magnetism  (§366) 
explains  the  structure  of  a  magnet  but  shifts  the  problem  of  the 
nature  of  "magnetism"  to  the  elementary  magnets.  Ampere 
suggested  that  the  molecule  of  a  magnetic  substance  is  a  magnet 
because  it  has  an  electric  current  flowing  about  it;  as  we  have 
already  seen  (§427)  a  circuit  in  which  a  current  flows  has  N  and 
S  polarity  like  a  magnet.  "According  to  Ampere's  theory," 
says  Maxwell,  "all  the  phenomena  of  magnetism  are  due  to 
electric  currents,  and  if  we  could  make  observations  of  the 
magnetic  force  in  the  interior  of  a  magnetic  molecule,  we  should 
find  that  it  obeyed  exactly  the  same  laws  as  the  force  in  a  region 
surrounded  by  any  other  electric  circuit."  That  is,  Ampere's 
theory  makes  magnetism  a  section  of  electrokinetics. 

How  the  elementary  electric  current  started  and  how  it  can 
continue  to  flow  about  the  "molecule"  without  consuming 
energy,  were  not  explained  by  Ampere.  In  the  electron  theory  of 
magnetism  which  is  an  extension  of  Ampere's  theory,  the  electric 
current  is  explained  as  due  to  electrons  or  corpuscles  which 
revolve  about  the  atom.  The  magnetization  of  the  elementary 
magnet  is  by  this  theory  due  to  the  magnetic  action  of  the 
revolving  electrons. 

According  to  the  electron  theory  of  matter,  one  or  more  electrons  are 
revolving  about  every  atom.  If  the  number  of  electrons  revolving  clock- 
wise is  equal  to  the  number  revolving  anti-clockwise,  then  the  atom  is 


ELECTROMAGNETS  AND  MAGNETIC  INDUCTION         435 


non-magnetic.  But  if  such  an  atom  is  brought  into  a  magnetic  field,  the 
effect  is  the  same  as  bringing  a  closed  circuit  into  the  field,  and  in  accordance 
with  Lena's  law  (§501),  magnetic  lines  are  set  up  which  are  opposite  to 
those  of  the  field  and  repulsion  results.  The  atom  is  thus  diamagnetic. 
The  character  of  the  electronic  orbit  is  changed  in  this  case  by  addition 
of  another  motion.  If  the  number  of  electrons  having  one  direction  of 
rotation  is  greater  than  the  number  having  the  opposite  direction,  then  the 
atom  is  naturally  para-magnetic.  In  the  case  of  para-magnetic  atoms,  it  is 
usually  assumed  that  not  only  are  all  the  velocities  in  the  electronic  orbits 
changed,  but  also  the  planes  of  the  orbits  and  probably  the  atoms  them- 
selves are  rotated  by  the  external  field. 

While  the  electron  theory  of  magnetism  is  the  most  promising  of  present 
theories,  it  is  not  developed  in  a  form  to  explain  all  the  facts. 

The  discovery  by  Heusler  of  a  strongly  ferro-magnetic  alloy  of  manganese, 
copper  and  aluminum,  the  component  parts  of  which  are  practically  non- 
magnetic, has  introduced  new  questions  as  to  the  nature  of  magnetism. 
As  these  ferro-magnetic  alloys  all  contain  manganese,  or  the  related  element 
chromium,  it  seems  probable  that  manganese  and  chromium  become 
ferro-magnetic  under  eertain  conditions  of  temperature  and  combination. 

497.  Residual  Magnetic  Effects. 
Hysteresis. — It  is  well  known  that 
hardened  steel  retains  its  magnetization, 
not  only  after  the  magnetizing  force  is 
removed,  but  also  against  opposing 
magnetic  fields  of  considerable  strength. 
Even  soft  iron  shows  small  magnetic 
effects  of  the  same  kind.  These  after- 
effects are  due  to  the  resistance  or  "re- 
luctanee"  which  the  molecular  magnets 
experience  to  free  rotation.  To  over- 
come this  resistance  to  changes  of 
magnetization,  work  is  required.  In  the  case  of  iron  subjected 
to  thousands  of  magnetic  reversals  in  a  minute,  as  in  the  alternat- 
ing current  transformer,  the  energy  expended  in  the  cyclic 
changes  of  magnetization  may  become  a  large  quantity. 

To  study  and  measure  these  phenomena  of  magnetic  lag,  the 
magnetization  curve  for  a  complete  cycle  is  made.  Starting 
with  non-magnetized  iron,  it  is  magnetized  by  an  increasing 
magnetic  field  until  it  approaches  saturation.  This  gives  us 
the  curve  Oabc  sometimes  called  the  "virgin  curve."  Then  the 
magnetizing  field  is  gradually  reduced  to  zero,  giving  the 
curve  cd,  which  shows  that  for  zero  magnetizing  field  Ht  there 


FIG.  361. 


436  ELECTRICITY  AND  MAGNETISM 

is  remanent  magnetism  represented  by  Od.  This  gives  a  measure 
of  the  retentivity  or  "remanence."  The  magnetizing  force  H 
is  now  reversed,  and  for  a  value  —  H  =  Oe,  the  iron  is  demag- 
netized. This  value  of  —  H  measures  the  coercive  force  of  the 
meal.  As  —H  is  increased  to  a  maximum,  decreased,  reversed, 
and  -|-  H  increased  in  the  original  direction,  the  parts  of  the 
curve  through  /,  g  and  h  are  traced,  until  the  loop  is  completed 
at  c.  There  is  thus  shown  to  be  a  lagging  of  the  magnetization 
behind  the  magnetizing  force  during  the  cycle. 

In  the  above  process  the  elementary  magnets  have  undergone 
a  double  reversal  in  direction,  and  work  has  been  done  against 
what  is  sometimes  called  "magnetic  friction."  Ewing  has 
shown  by  his  model  (§366)  that  this  "magnetic  friction"  can 
be  explained  by  the  mutual  actions  of  the  elementary  magnets. 
The  phenomenon  is  called  hysteresis.  The  hysteresis  loss  is 
stated  in  ergs  per  cu.  cm.  per  cycle  and*  is  proportional  to  the 
area  enclosed  by  the  hysteresis  loop.  This  energy  is  dissipated 
in  heating  the  iron.  It  should  be  noted  that  this  is  entirely 
different  from  the  heating  due  to  induced  eddy  currents  in  the 
iron  (§507). 

Iron  varies  greatly  in  its  quality  as  regards  hysteresis.  Thus  Ewing 
found  for  a  certain  soft  iron  at  a  maximum  induction  of  5000  lines,  a  loss 
of  910  ergs  per  cu.  cm.  per  cycle  and  at  an  induction  of  9000  a  loss  of  2300 
ergs  per  cycle.  For  cast  iron  at  the  same  maximum  induction  the  loss 
may  reach  a  value  of  ten  times  the  above.  It  may  be  added  that  a 
loss  of  2300  ergs  per  cu.  cm.  per  cycle  represents  1.36  watts  per  pound 
of  iron  per  100  cycles  per  second,  or  over  three  and  a  half  H.  P.  per  ton 
of  iron. 

498.  Energy  of  Magnetic  Field. — A  magnetic  field  contains  energy  in  the 
form  of  strains  of  some  kind  in  the  "ether."  The  amount  of  this  energy 
per  cubic  centimeter  is  calculated  as  in  the  case  of  an  electrostatic  field 
(§419),  and  is  given  by  the  similar  expression  energy  E  =  nH*/8K,  where  E 
is  in  ergs,  H  is  the  intensity  of  the  field  and  fi  is  the  permeability  of  t  he  medium . 
The  importance  of  this  energy  of  the  magnetic  field  appears  in  electro- 
magnetic induction.  In  the  case  of  the  alternating  transformer  (§520),  the 
energy  of  the  primary  circuit  is  converted  into  the  energy  of  the  magnetic 
field  and  this  magnetic  energy  is  then  transformed  into  the  electrical  energy 
of  the  secondary  circuit.  .In  the  impedance  or  choking  coil  (§523),  the 
energy  of  the  circuit  is  transformed  into  the  energy  of  the  magnetic  field, 
and  then  transformed  back  into  the  energy  of  the  electric  circuit  as  many 
times  a  second  as  there  are  alternations. 


ELECTROMAGNETIC  INDUCTION  437 

ELECTROMAGNETIC  INDUCTION 

499.  Induced  Electric  Currents.— On  November  24,  1831, 
Michael  Faraday  described  to  the  Royal  Society  of  London  a 
series  of  experiments  showing  that  electric  currents  can  be 
produced  in  a  closed  conducting  circuit,  (a)  by  moving  neigh- 
boring magnets;  or  (6)  by  changing  the  current  in  a  neighbor- 
ing electric  circuit;  or  (c)  by  moving  a  neighboring  electric 
circuit.  An  electric  current  thus  produced  is  said  to  be  induced, 
and  the  phenomenon  is  called  electromagnetic  induction.  Few 
discoveries  in  science  have  had  such  important  practical  results 
as  this  discovery  of  Faraday's.  Almost  every  modern  indus- 
trial application  of  electricity  depends  upon  electromagnetic 
induction.1 

600.  Faraday's  Experiments. — The  experiments  on  induced 
currents  made  by  Faraday  were  the  following:  (I)  A  coil  of 
wire  B  forms  a  closed  circuit  through  a  sensitive  galvanometer 
0  (Fig.  362).  When  the  pole  of  a  magnet  is  brought  up  to  Bt  a 
momentary  current  is  induced  in  B,  and  the  galvanometer  needle 
is  deflected.  When  the  magnet  pole  is  removed,  a  momentary 
current  is  again  induced,  but  in  the  opposite  direction  to  that 
upon  approach.  The  following  facts  may  be  noted:  (a)  The 
essential  motion  is  relative,  that  is,  moving  the  coil  to  or  from 
the  magnet  produces  the  same  effect  as  moving  the  magnet; 
(b)  the  current  lasts  only  during  the  time  of  motion;  when  the 
magnet  and  the  coil  are  relatively 
at  rest,  there  is  no  induced  cur- 
rent; (c)  bringing  up  a  N  pole 
to  a  coil  induces  a  current  anti- 
clockwise as  seen  from  the  pole; 
that  is,  the  induced  current 
makes  this  face  a  N  face  (§427). 
Thus  the  approaching  N  pole  is 
repelled  by  the  magnetic  action 

of  the  induced  current.     Removing  the  N  pole  induces  a  clock- 
wise current,  that  is,  makes  the  coil  face  a  S  face.     Thus  the  N 

1  Working  at  the  same  time,  Joseph  Henry  discovered  independently  the  fundamental  1 
facts  of  electromagnetic  induction,  and  probably  even  anticipated  Faraday  in  some  cases, 
But  Professor  Henry  worked  under  many  disadvantages  in  the  isolated  town  of  Albany. 
New  York,  and  his  discoveries  were  not  widely  known  at  the  time  they  were  made. 


438 


ELECTRICITY  AND  MAGNETISM 


Fia.  363. 


pole  is  attracted  as  it  is  removed.  An  approaching  S  pole 
induces  a  current  in  the  same  direction  as  a  receding  N  pole,  and 
vice  versa  (Fig.  363).  Or  in  general,  the  mag- 
netic action  of  the  induced  current  opposes  the 
motion  of  the  magnet.  This  is  evidently  a 
case  of  action  and  reaction.  If  the  approach- 
ing magnet  were  attracted  by  the  induced 
current,  it  would  require  no  work  to  bring  the 
magnet  up,  and  we  would  get  an  electric  cur- 
rent, which  represents  energy,  without  the 
expenditure  of  work.  This  would  be  contrary 
to  the  principle  of  the  conservation  of  energy 
(§341). 

We  can  now  describe  the  above  experiment 
in  the  convenient  terms  of  the  magnetic  field 
and  the  magnetic  lines  of  force,  as  conceived 
by  Faraday  (§368).  The  magnet  is  sur- 
rounded by  a  magnetic  field,  and  the  lines  of 
force  emerge  from  the  N  pole,  and  enter  at  the 
S  pole.  The  motion  of  the  magnet  thus 
changes  the  number  of  lines  of  force  included  by  the  coil.  The 
experiment  thus  shows  that,  (a)  a  change  of  the  number  of  lines 
of  magnetic  force  included  by  a  circuit  induces  a  current  in  the 
circuit;  (6)  the  current  induced  is  proportional  to  the  rate  of 
change  of  included  lines  of  force;  and  (c)  the  magnetic  lines  from 
the  induced  current  increase  as  the  magnetic  lines  from  the 
magnet  decrease  through  the  circuit,  and  vice  versa.  The 
positive  direction  of  change  of  lines  is  to  be  reckoned  in  the 
same  direction  for  both  magnet  and  current.  Faraday's  other 
experiments  are  now  easily  described. 

(II)  Substitute  for  the  magnet  NS,  a  coil  carrying  an  electric 
current.  A  is  thus  surrounded  by  a  magnetic  field  (§427),  and 
moving  A  in  front  of  B,  changes  the  number  of  magnetic  lines 
included  by  B,  and  thus  induces  an  electric  current  in  B  during 
the  time  of  motion.  When  A  is  approaching  B,  the  opposing 
faces  of  the  two  coils  are  either  both  N  or  both  S;  and  for  this 
case  the  induced  current  in  B  must  be  inverse  in  direction  to  the 
current  in  A.  Similarly  it  is  seen  that  upon  the  receding  of  A, 
the  induced  current  is  direct  to  that  in  A. 


ELECTROMAGNETIC  INDUCTION 


439 


The  coil  A  carrying  the  original  or  inducing  current,  is  called 
the  primary  coil  (Pr) ,  and  its  current  the  primary  current.  The 
coil  B  is  called  the  secondary  coil  (Sc),  and  the  induced  current, 
the  secondary  current.  A 

current  in  the  same  direction  A          B 

as  the  primary  current  is 
called  direct,  and  a  current 
in  the  opposite  direction  is 
called  inverse. 

(Ill)  With  two  coils  A  and 
B  as  before,  we  can  change  FIO.  364. 

the  number  of  lines  of  force 

through  B,  by  changing  the  current  in  A  (Fig.  364).  Thus  we 
find  that,  making  or  increasing  the  Pr  current  induces  a 
momentary  inverse  current  Sc  in  B;  and  that,  breaking  or  de- 
creasing the  Pr  current  induces  a  direct  current  in  the  secondary 
circuit. 


FIG    366. 

The  coils  A  and  B  may  be  placed  anywhere,  provided  that  the 
magnetic  lines  from  A  pass  through  B.  Coils  wound  over  or 
alongside  of  each  other  on  a  cylinder  are  common  arrangements 
for  obtaining  maximum  induced  currents  (Fig.  365).  Two 
straight  parallel  wires  may  similarly  form  primary  and  secondary 
circuits.  This  is  often  the  case  in  telephone  circuits,  and  is  the 
cause  of  the  "cross  talking"  in  the  lines. 

(IV)  If  A  and  B,  one  or  both,  have  an  iron  core,  the  induced 
currents  are  greater,  but  in  the  same  direction  as  without  the 
iron  cores.  This  is  easily  explained  in  terms  of  the  magnetic 
lines.  Iron  has  a  greater  magnetic  permeability  (§491)  than 
air,  so  that  a  given  change  in  the  primary  current  produces  a 
greater  change  in  the  magnetic  flux  through  the  secondary  circuit, 
and  thus  causes  a  greater  induced  current.  Figs.  365  and  366 


440 


ELECTRICITY  AND  MAGNETISM 


how  common  arrangements  of  the  primary  and  secondary  coils 
on  iron  cores.  The  arrangement  shown  in  Fig.  366  is  one  Fara- 
day used  in  his  earliest  experiments. 

501.  Lenz's  Law. — In  1834,  Lenz  stated  the  following  impor- 
tant relation  between  the  induced  current  and  the  motion  of  the 
electrical  circuit  or  the  magnet  causing  the  induction:  The  in- 
duced current  is  in  such  a  direction  as  to  oppose  by  its  electromagnetic 
action  the  motion  of  the  magnet  or  the  coil  which  produces  the 
induction.  We  have  already  seen  that  this  holds  in  the  case  of 
the  motion  of  a  magnet/  and  it  is  easily  seen  that  it  also  holds 
for  two  coils  moving  relatively  to  each  other. 


Fio    367a. 


Fia.  3676. 


Lenz's  law  can  be  extended  to  the  case  of  a  current  induced 
by  the  variation  of  a  primary  current,  but  the  reactions  are  then 
purely  electromagnetic.  When  the  current  is  induced  in  a 
secondary  coil  B  by  making  or  breaking  the  current  in  a  primary 
coil  A,  we  have  a  reaction  of  B  on  the  current  in  A.  That  is  the 
electromagnetic  induction  is  mutual.  We  thus  have  the  follow- 
ing series  of  actions  and  reactions:  Starting  a  current  in  A 
produces  magnetic  lines  through  A,  and  part  of  them  pass 
through  B.  There  is  thus  an  inverse  current  induced  in  B.  But 
this  induced  current  started  in  B  produces  lines  which  are  oppo- 
site to  those  produced  by  the  primary  current  in  A.  The  effect 
of  the  induced  current  is  thus  to  oppose  and  retard  the  building 
up  of  the  magnetic  field  through  the  coils. 

Upon  breaking  the  primary  current,  the  current  induced  in  B 
is  direct;  that  is,  this  induced  current  produces  magnetic  lines 
which  are  in  the  same  direction  as  those  produced  by  A.  The 


ELECTROMAGNETIC  INDUCTION  441 

effect  of  the  induced  current  in  B  is  thus  to  maintain  the  field, 
that  is,  to  delay  the  decrease  of  the  number  of  lines  of  force 
through  the  coils. 

It  can  thus  be  seen  that  in  general,  the  magnetic  action  of  the  in- 
duced current  opposes  the  magnetic  action  of  the  inducing  current. 

The  study  of  the  actions  and  reactions  of  the  primary  and  sec- 
ondary currents,  with  their  energy  relations,  is  very  important 
in  understanding  the  complete  theory  of  the  induction  coil  (§511), 
and  of  the  alternating  current  transformer  (§520). 

502.  Induction  by  Cutting  Lines  of  Force. — One  of  Faraday's 
early  observations  was  that  "single  wires,  approximated  in  certain 
directions  toward  the  magnetic  pole  (of  a  large  electromagnet), 
had  currents  induced  in  them."  It  is  often  convenient  to  con- 
sider the  induction  as  due  to  the  motion  of  a  single  wire  across 
lines  of  force,  or  as  we  often  say,  due  to  "cutting  lines  of  force." 


Fio.  3686. 

Thus  suppose  the  wire  A  B  moved  across  the  field  between  the 
poles  N  and  S  (e.g.,  of  a  U-shaped  magnet).  In  case  (a)  the 
wire  forms  part  of  a  complete  circuit  ABG.  In  moving  AB  down, 
the  number  of  lines  through  the  circuit  is  increased,  and  an 
anti-clockwise  current  (seen  from  below)  is  induced,  that  is  from 
A  to  B  in  the  part  AB  of  the  circuit.  In  case  (6),  the  wire  forms 
part  of  the  circuit  ABG' ,  and  the  motion  downward  induces  a 
current  in^the  circuit,  so  that  the  current  flows  from  A  to  B. 
The  current  is  clockwise  as  seen  from  below.  When  the  motion 
is  upward,  the  current  is  evidently  from  B  to  A.  The  three 
directions,  of  magnetic  field,  motion,  and  induced  current,  are 
thus  mutually  at  right  angles,  as  indicated  in  the  rectangular 
axes  of  Fig.  368a.  Professor  J.  A.  Fleming  has  given  a  conven- 
ient rule  for  remembering  these  relative  directions.  Holding 
the  right  hand  as  indicated  in  Fig.  3686,  with  the  thumb,  the  fore- 
finger and  the  center  finger,  making  right  angles  with  each  other, 
then  if  the  forefinger  is  held  in  the  direction  of  the  magnetic 


442 


ELECTRICITY  AND  MAGNETISM 


field,  and  the  thumb  in  the  direction  of  the  motion,  the  center 
finger  will  indicate  the  direction  of  the  current. 

503.  Numerical  Calculation  of  Induced  E.M.F. — By  Ohm's  law,  the 
induced  current  varies  directly  as  the  induced  electromotive  force  and 
inversely  as  the  resistance  of  the  circuit.  The  resistance  is  a  constant  for 
the  circuit  (§442).  Experiments  show  that  the  electromotive  force 
induced  in  a  circuit  is  proportional  to  the  rate  of  variation  of  lines  of  force 
through  the  circuit,  that  is, 


I 


where  i  is  the  time  in  seconds,  and  Nl  and  JV,  are  the  number  of  lines  at 

the  beginning,  and  at  the  end  of  the 
time  interval  t.  If  the  variation  of  N 
is  uniform  during  the  time  t,  then  the 
e.m.f.  induced  is  constant.  When  the 
variation  is  not  uniform  the  e.m.f.  at 
any  instant  is  given  by  the  differential 
coefficient,  that  is,  E=-KdN/dt. 
In  these  expressions  K  is  a  constant. 
If  E,  N  and  t  are  expressed  in  c.  g.  s. 

units,  then  it  can  be  shown  that  K  becomes  unity  (§504).     To  express  E 

in  volts,  we  divide  by  108  (§441),  that  is, 


I  Positive 

•*  \J 


FIQ.  369. 


108/          10"  dt 

The  negative  sign  is  explained  as  follows:  The  positive  direction  of  the 
lines  of  force  is  taken  as  that  of  the  advance  or  thrust  of  a  right-Kanded 
screw,  Fig.  369,  where  the  rotation  of  the  same  gives  the  positive  direction 
of  the  e.m.f.,  or  current.  Thus  a  positive  increase  of  N  corresponds  to 
a  negative  induced  e.m.f. 

504.  A  Second  Numerical  Statement  of  Induced  E.M.F. — It  is  often 
convenient  to  calculate  the  induced  e.m.f.  in  terms  of  the  number  of  lines 
of  force  cut  by  a  conductor  per  second.  Let  H  •»  the 
strength  of  the  magnetic  field,  =the  number  of  lines  of 
force  per  square  centimeter  section  of  the  field  (§374), 
and  let  J  =  the  length  of  the  conductor  in  centimeters, 
and  t>=«the  velocity  in  centimeters  per  second.  If  the 
motion  is  perpendicular  to  the  lines  of  force  and  also 
to  the  length  direction  of  the  conductor,  then  the 
number  of  lines  cut  per  second  is  IvH,  or  the  induced 
e.m.f.  is 

IvH 


Fia.  370. 


E=lvH,  or^(volts) 


108 


In  case  the  velocity  v  and  the  conductor  length  are  not  at  right  angles  to 
the  field  Ht  their  components  at  right  angles  to  H  are  to  be  taken. 

We  can  derive  the  above  equation  by  equating  the  mechanical  work  done 
in  moving  the  conductors  across  the  magnetic  field  to  the  electrical  energy 


ELECTROMAGNETIC  INDUCTION 


443 


of  the  induced  current.  If  *  is  the  induced  current,  the  electrical  energy 
produced  in  time  t  is  W*=Eit  (§441).  In  §529  it  is  proved  that  a  force 
F~Hil  acts  on  the  conductor,  and  that  this  force  is  in  the  opposite  direction 
to  the  motion  which  induces  the  current.  The  distance  moved  by  the  con- 
ductor in  the  time  t  is  d  =  vt.  Hence  the  mechanical  work  is  W^Fd** 
Hilvt.  Equating  the  electrical  energy  to  this,  we  get  E  =  Hvl.  Hence 
the  constant  K  in  the  equation  of  §503  must  be  unity  for  the  absolute 
c.g.s.  units. 

606.  Calculations   for   Current   and   Electric    Quantity. — In   the  above 
section  (§503)  it  has  been  shown  that  the  induced  e.m.f. 
.,        Nt-N,        dN 

E=    —    ~ar 

The  current  is  then 


.     _ 
~ 


Rt  Rdt 

Thus  we  have  lt=  -(N3-NJ/R  and  Idt^-dN/R.  But  It  =  Q,  the 
total  flow,  of  electric  quantity  in  the  time  t,  and  Idt  =  dQ,  the  electric  quantity 
in  the  time  dt.  Thus  the  total  quantity  of  electricity  induced  is 


R 


Nv         CdN 
~J    R 


This  quantity  can  be  measured  by  the  throw  of  a  ballistic  galvanometer, 
when  the  time  of  induction  is  a  small  fraction  of  the  period  of  the  galvano- 
meter needle  (§439).  The  above  relations  are  in  absolute  units. 

606.  Faraday's  Disk  Dynamo.  —  One  of  Faraday's  earliest  ex- 
periments in  electromagnetic  induction  was  to  rotate  a  copper 
disk  between  the  poles  of  a  magnet,  the  plane  of  the  disk  being 
perpendicular  to  the  field  (Fig. 
371).  A  galvanometer  .circuit 
was  completed  by  wires  sliding 
on  the  axle  and  on  the  circum- 
ference of  the  disk,  and  a  current 
was  '  shown  by  the  deflection  of 
the  galvanometer  during  the  rota- 
tion of  the  disk.  In  this  ma- 
chine each  radius  of  the  disk  cuts 
the  lines  of  the  field  at  the  rate  of 
nr^nH  per  second,  where  nr2  is  the 
area  of  the  disk,  H  is  the  strength 
of  the  field  assumed  uniform,  and 
n  is  the  number  of  revolutions  per  second  of  the  disk.  Thus 

7-T/         V      \ 

E  (volts)  = 


FIQ.  371. 


444 


ELECTRICITY  AND  MAGNETISM 


Fia.  372. 


This  arrangement  made  the  first  dynamo-electric  machine. 
Forbes  and  others  have  attempted  to  use  this  as  a  model  for 
commercial  electric  generators,  but  the  e.m.f.,  with  any  practical 
diameters  and  speeds  is  too  small  for  industrial  uses. 

507.  Foucault  or  Eddy  Currents. — It  was  observed  a  number 
of  years  before  Faraday's  discovery  of  induced  currents,  that  a 

vibrating  magnetic  needle  quickly  came 
to  rest  when  near  or  over  a  copper  plate. 
Arago  had  in  1824  also  shown  that  a 
magnetic  needle  suspended  over  a  rotat- 
ing copper  disk,  rotates  with  the  disk 
(Fig.  372).  Both  the  damping  of  the 
needle,  and  Arago's  disk  experiment 
were  explained  by  Faraday  as  phenomena 
of  electromagnetic  induction.  The  rela- 
tive motion  of  the  magnet  and  the  disk 

induces  an  e.m.f.  in  the  metal  disk.  The  current  thus  generated 
circulates  in  the  disk,  producing  a  magnetic  action,  which  by 
Lenz's  law  tends  to  hold  the  magnet  at  rest  relative  to  the  disk 
or  plate. 

Electric  currents,  thus  induced  and  circulating  in  a  metal  mass, 
are-  called  eddy  currents  or  Foucault  currents.  The  energy  of 
such  currents  is  dissipated  in  heat.  The  iron  cores  of  armatures 
of  dynamo  machines  and  transformers  are  always  laminated  so  as 
to  offer  very  high  resistance  to  the  formation  of  such  currents, 
and  thus  to  stop  the  heat  losses. 

An  interesting  example  of  damping  by  eddy 
currents  is  shown  in  Fig.  373.  The  pendulum 
with  its  copper  plate  swings  freely  if  the  electro- 
magnet is  not  excited,  but  is  damped  immediately 
when  the  magnetic  field  is  made. 

This  damping  action  of  eddy  currents  is  often 
taken  advantage  of  in  d'Arsonval  or  movable  coil 
galvanometers  (§438),  to  bring  the  moving  coil  to 
rest  quickly.  The  coil  is  wound  on  a  closed  copper 
frame  in  which  the  eddy  currents  are  generated  during  the 
vibration.  The  coil  itself  is  damped  in  the  same  way  on  closed 
circuit.  The  magnetic  needles  of  galvanometers  are  also  often 
damped  by  suspending  them  in  openings  in  copper  blocks. 


Fio.  373. 


ELECTROMAGNETIC  INDUCTION  445 

508.  Self-induction. — Several  persons  seem  to  have  observed 
independently  about  1832,  that  there  is  a  bright  spark  when  a 
current  is  broken  in  a  circuit  containing  an  electro-magnet.  On 
making  a  current  in  the  same  circuit,  there  is  no  such  spark. 
This  extra  current  at  break  was  noticed  by  persons  receiving  a 
shock  upon  breaking  the  circuit  of  an  electromagnet,  if  they  had 
the  terminals  in  their  hands  at  the  time  of  break.  In  investi- 
gating this,  Faraday  observed  the  following  facts.  Upon  break- 
ing the  circuit  of  a  helix  without  an  iron  core,  a  similar  bright 
spark  is"  obtained,  only  less  than  in  the  case  of  an  electromagnet. 
Likewise,  when  the  current  in  a  long  straight  wire  was  broken, 
a  spark  occurred,  only  less  bright  than  in  the  case  of  the  helix. 
Upon  breaking  a  current  in  a  short  wire  there  was  practically  no 
spark.  Also  in  the  case  of  a  long  wire  doubled  back  on  itself, 
there  was  no  extra  spark  at  break.  "  The  first  thought  that  arises 
in  the  mind, "  wrote  Faraday, "  is  that  the  electricity  circulates  with 
something  like  momentum,  or  inertia  in  the  wire,  and  that  thus, 
a  long  wire  produces  effects  at  the  instant  the  current  is  stopped, 
which  a  short  wire  cannot  produce.  Such  an  explanation 
is  however  at  once  set  aside  by  the  fact  that  the  same  length  of 
wire  produces  the  same  effects  in  very  different  degrees,  according 
as  it  is  simply  extended,  or  made  into  a  helix,  or  forms  the  circuit 
of  an  electromagnet."  Faraday  then  proceeded  to  show  that  this 
extra  current  was  due  to  electromagnetic  induction,  from  the 
varying  current  acting  on  its  own  circuit.  This  phenomenon  of  a 
current  inducing  an  extra  electromotive  force  in  its  own  circuit, 
is  called  self-induction.  A  circuit  includes  in  general  lines  of 
force  due  to  its  own  current.  Breaking  the  current  thus  removes 
the  lines  of  force,  or  has  the  same  effect  as  removing  a  magnet. 
Suppose  the  current  flows  clockwise  in  a  circular  circuit;  break- 
ing the  current  then  removes  positive  lines,  or  is  the  same  as 
removing  a  north  pole.  But  this  induces  a  clockwise  e.m.f., 
that  is  a  direct  current;  this  adds  itself  to  the  current  being 
broken,  and  thus  causes  the  bright  spark  of  break.  The  effect  of 
an  iron  core  is  to  increase  the  magnetic  induction,  and  thus  to 
increase  the  e.m.f.  of  self-induction  at  break.  Self-induction 
thus  prolongs  the  current  at  break,  or  acts  to  retard  a  decrease 
of  the  current.  When  the  circuit  is  wound  back  on  itself,  so 
that  it  includes  no  lines  of  force,  there  can  be  no  change  of  lines 


446  ELECTRICITY  AND  MAGNETISM 

of  force,  and  hence  no  self-induction.  Such  circuits  are  said 
to  be  non-inductive  or  inductionless.  The  wire  in  resistance 
boxes  is  wound  doubled  from  its  middle  point  as  is  shown  in 
Fig.  321. 

When  a  current  is  made  or  increased  in  an  inductive  circuit, 
such  as  a  helix,  magnetic  lines  are  put  through  the  circuit.  Thus 
positive  lines  enter  at  the  face  in  which  the  current  is  clockwise 

(S  face),  and  this  is  equiva- 
lent to  bringing  up  a  N  pole. 
But  this  induces  an  anti-clock- 

Fio.  374. 

wise  e.m.f.,  that  is  an  e.m.f. 

inverse  to  the  starting  current.  That  is,  the  building  up  of  a 
current  in  a  coil,  is  accompanied  by  an  induced  inverse  e.  m.  f. 
at  the  time  of  the  current  increase.  Here  again  the  self-induc- 
tion opposes  and  delays  the  current  changes.  Fig.  374  shows 
the  growth  and  dying  away  of  currents  in  an  inductive  circuit 
as  observed  in  an  oscillograph  (§524).  Helmholtz  deduced  in 
1851  an  equation  showing  the  law  of  the  growth  of  currents  in 
inductive  circuits  and  these  oscillograph  curves  confirm  the 
Helmholtz  equation.1 

509.  Coefficient  of  Self-induction  or  Inductance.  —  The  e.m.f. 
of  self-induction  in  a  circuit  thus  depends  upon  the  change  in  the 
number  of  lines  of  force  through  the  circuit,  caused  by  the  varia- 
tion of  the  current.  The  number  of  lines  evidently  depends  upon 
(a)  the  current  /,  and  (6)  upon  the  dimensions  of  the  circuit, 
and  (c)  the  presence  of  magnetic  substances,  such  as  an  iron  core. 
For  a  circuit  without  iron,  N  the'  number  of  lines  included  by  the 
circuit  varies  directly  as  I,  or  N  =  LI,  where  L  is  the  coefficient 
of  self-induction  or  the  inductance  of  the  circuit.  Thus  the  induc- 
tance of  a  circuit  is  numerically  equal  to  the  increase  in  the  num- 

1  Helmholtz's  equation  is 


•where  I,  E,  R,  L  and  t  represent  the  quantities  indicated  in  this  and  the  next  section,  and 
0  is  the  base  of  the  Naperian  logarithms.  This  equation  is  deduced  as  follows.  The 
e.  m.  f.  in  the  circuit  at  any  instant  is  equal  to  the  impressed  e.  m.  f.  less  the  counter  e.  m,  f  . 
of  self-induction,  or  e—E—L  dl/dt.  Hence  the  current  is 

E  _  L  dl 
~~R       if  dF 
By  integration  of  this  differential  equation  we  get  the  above  equation  of  Helmholtn. 


ELECTROMAGNETIC  INDUCTION 


447 


ber  of  magnetic  lines  included  by  the  circuit  for  unit  increase  of 
the  current.  For  a  circuit  with  an  iron  core,  this  increase  of 
magnetic  lines  per  unit  current  is  not  con- 
stant, because  the  magnetic  permeability 
of  iron  varies  with  the  magnetizing  force 
(§491) .  Hence  L  the  inductance  of  a  circuit 
with  an  iron  core  is  a  variable  depending 
upon  the  magnetic  curve  of  the  iron. 

We  can  also  express  the  inductance  of  a 
circuit  in  terms  of  the  e.m.f.  induced  for 
unit  rate  of  change  of  the  current  in  the 
circuit.  This  is  shown  as  follows:  The 
number  of  magnetic  lines  through  a  circuit 
is  N  =  LI',  hence  the  induced  e.m.f.  is  E  — 
—  dN/dt=—L(d!/dt).  If  the  rate  of  change  of  the  current  is 
unity,  that  is  if  dl/dt  =  l,  then  E  =  L.  We  can  thus  define  unit 
inductance,  as  the  inductance  of  a  circuit  in  which  unit  e.m.f.  is 
induced  by  unit  change  of  current  per  second  in  the  circuit.  The 
practical  unit  of  inductance  is  the  henry  and  is  equal  to  10'  C.G.S 


Fio.  375. 


M 


Fia.  376. 


units  of  inductance.  The  henry  can  be  defined  as  the  inductance 
of  a  circuit,  in  which  a  change  of  one  ampere  per  second  produces 
an  induced  e.m.f.  of  one  volt.  Standards  of  inductance  are 
used  in  the  shape  of  coils  wound  on  marble  or  other  non-magnetic, 
and  permanent  cores.  These  are  graduated  in  multiples  or 
submultiples  of  the  henry.  A  variable  standard  of  inductance 
can  be  made  by  two  coils  joined  in  series  and  arranged  so  that 
they  can  be  rotated  in  reference  to  each  other,  and  thus  change 


448 


ELECTRICITY  AND  MAGNETISM 


Fio.  378. 


the   total   lines    included.     Such    an    inductance    standard    is 
illustrated  in  Fig.  375. 

510.  Experiments   on   Self-induction. — To   demonstrate    extra-currents 
due  to  self-induction,  Faraday  made  the  following  experiment:     In  a  circuit 
containing  a  large  helix  or  electromagnet  Af,  there  is  a  galvanometer  G, 
the  galvanometer  being  in  parallel  with  the  helix  M  (Fig.  376).     The  current 
from  the  battery  B  is  made  or  broken  by  the  key  K.     In  the  steady  con- 
dition, the  current  divides  between  the  helix  and  the  galvanometer,  and  there 

is  a  steady  deflection  of  the  galvanometer 
needle,  say  of  n  degrees  to  the  right.  A 
stop  is  placed  so  that  the  needle  can  not 
deflect  to  the  right.  Upon  breaking  the 
current  at  K,  there  is  a  throw  of  the 
galvanometer  to  the  left,  due  to  the  extra- 
current  of  break  flowing  back  through,  the 
galvanometer  circuit.  Evidently  the  extra- 
current  in  M  is  in  the  same  direction  as  the 
current  being  broken. 

A  striking  variation  of  the  above  experi- 
ment is  to  put  an  incandescent  lamp  in 
parallel  with  an  electromagnet,  passing  just  enough  current  to  bring  the 
lamp  to  a  red  glow  (Fig.  377).  Upon  breaking  the  current,  the  lamp 
flashes  brightly  ror  an  instant,  due  to  the  e.m.f.  of  self-induction  at  break. 

The  best  way  of  showing  the  effects  of  self-induction  is  by  the  use  of  the 
Wheatstone  bridge,  as  described  by  Maxwell.  In  the  bridge  arrangement, 
Fig.  378,  the  resistances  Ru  Rt  and  R9  are  wound  non-inductively  ({508), 
and  Rm  is  the  resistance  of  an  electromagnet.  The  resistances  are  arranged 
so  that  R!'.  R3  =  Rt:  Rm.  There  is  accordingly  no  deflection  of  the  galva- 
nometer G  when  the  current  is  in  a  steady  state.  But  upon  making  the 
current  by  closing  the  key  K,  there  is  a  momentary  throw  of  the  galva- 
nometer needle,  the  deflection  of  the  needle  being  again  zero  when  the  current 
reaches  a  steady  state.  Upon  breaking  the  current,  there  is  a  throw  of  the 
needle  in  the  opposite  direction  to  that  at  make.  If  all  four  of  the  resistances 
are  non-inductive,  there  are  no  such  momentary  throws  of  the  galva- 
nometer needle  at  make  and  break.  Suppose  the  current  enters  at  c;  then 
the  current  reaches  its  full  value  in  cRtb  sooner  than  in  cRmd,  so  that  there 
will  be  a  deflection  of  the  galvanometer  showing  a  momentary  current  from 
b  to  d.  Upon  break  the  momentary  flow  will  be  from  d  to  6. 

The  explanation  of  this  is  evident  from  §508.  The  above  method  of 
showing  the  extra-currents  has  been  developed  by  Maxwell  and  others  into 
a  method  of  measuring  the  coefficient  of  self-induction.  For  these  methods 
the  reader  must  refer  to  the  laboratory  manuals. 

511.  The  Induction  Coil. — The  induction  coil  is  a  piece  of  appa- 
ratus for  producing  pulsating  currents  or  discharges  of  high 
e,m.f.  in  a  secondary  circuit,  by  making  and  breaking  a  current 


ELECTROMAGNETIC  INDUCTION 


449 


Fio.  379. 


in  a  primary  circuit.  The  current  in  the  primary  circuit  may  be 
from  a  battery  with  only  a  few  volts  e.m.f.  In  Fig.  379,  we  have 
a  diagram  showing  the  parts  and  ar- 
rangement of  the  ordinary  induction 
coil.  The  primary  circuit  Pr  consists 
of  (a)  a  solenoidal  coil  P  with  a  bundle 
of  soft  iron  wires  as  core;  (b)  an  inter- 
rupter K  for  making  and  breaking  the 
primary  current.  When  the  inter- 
rupter is  mechanical  as  shown  in  the 
figure,  there  is  a  condenser  joined 
across  the  gap  to  lessen  the  extra  spark  of  break,  and  thus 
cause  a  quicker  break  of  the  current;  (c)  the  secondary  circuit 
Sc  consisting  of  a  solenoidal  coil  S  surrounding  the  coil  P,  and  a 
spark  gap  D.  The  secondary  coil  is  wound  with  many  turns 
of  fine  wire.  To  increase  the  insulation,  this  coil  is  also  wound 
in  disk  sections.  The  primary  coil  is  wound  with  a  compara- 
tively few  turns  of  much  larger  wire. 

Making  the  primary  current  produces  magnetic  lines  which 
thread  through  the  secondary.  These  lines  are  removed  upon 
breaking  the  primary  current.  Thus  there  is  induced  in  the 
coil  S  an  inverse  e.m.f.  at  make,  and  a  direct  e.m.f.  at  break  of 
the  primary  current.  The  break  in  most  coils  is  much  quicker 
than  the  make,  and  thus  the  direct  induced  e.m.f.  in  Sc  is  so 
much  greater  than  the  inverse  induced  e.m.f.,  that  the  discharge 
effects  are  mostly  uni-directional.  The  reason  for  this  is  that 
the  growth  of  the  primary  current  at  make  is  retarded  by  the 
inductance  of  the  circuit  (§508),  while  with  .a  good  interrupter 
and  proper  condenser,  the  break  can  be  made  very  sharp.  The 
greater  the  number  of  turns  of  the  secondary  coil, 
the  greater  the  induced  e.m.f.  The  resistance  of 
the  coil  is  of  course  high,  and  consequently  the 
current  small. 

In  small  induction  coils  the  Wagner  hammer  is 
the  common  form  of  interrupter.  This  is  shown 
at  K  in  Fig.  379,  and  its  action  can  be  easily 
seen.  In  large  coils,  a  form  often  used  consists 
of  a  brush  sliding  on  a  revolving  commutator  driven  by  an 
electric  motor.  The  electrolytic  interrupter  of  Wehnelt  is  also 

29 


Fio.  380. 


450 


ELECTRICITY  AND  MAGNETISM 


Fio.  381. 


frequently  used  (see  Fig.  380).  P  is  a  platinum  wire  in  a  solu- 
tion of  sulphuric  acid,  L  is  a  lead  plate.  Only  the  point  of  the 
wire  is  exposed  to  the  acid.  When  P  is  made  the  anode,  and  L 
the  kathode,  gas  forms  at  P,  interrupts  the  current,  and  escapes 
in  bubbles,  and  thus  the  current  is  again  made. 

512.  The  Tesla  Induction  Coil. — To 
obtain  currents  of  very  high  fre- 
quencies and  high  electromotive 
forces,  Tesla  used  a  form  of  induction 
coil  in  which  the  oscillatory  discharge 
of  a  Leyden  jar  (§541)  is  used  as  in- 
terrupter. The  terminals  of  the 
secondary  of  an  induction  coil  7  (Fig. 
381),  are  connected,  one  to  the  inner 
coating,  and  one  to  the  outer  coating 

of  an  insulated  Leyden  jar  C.  The  circuit  is  completed 
through  the  primary  winding  of  the  Tesla  coil,  and  the 
discharge  balls.  The  primary  of  the  Tesla  coil  A,  consists 
of  a  half  dozen  turns  of  wires  on  a  non-magnetic  core.  The 
coils  A  and  B  are  separated  by  air  or  oil  as  insulation.  The 
alternations  at  D  from  the  Leyden  jar  may  have  a  frequency 
of  several  millions  per  second  (§541).  Hence  the  currents 
induced  in  B  are  not  only  of  high  e.m.f.  but  also  of  very  high 
frequency. 

513.  Electromotive  Force  in  a  Coil  Rotating  in  a  Magnetic 
Field. — In  the  earth  inductor  and  in  many  dynamo-electric 
machines,  electric  currents  are  produced  by  rotating  coils  of  wire 
in  a  magnetic  field.  We  take  the  simple  case  of  a  rectangular 
coil  in  a  uniform  magnetic  field.  The  coil  A  BCD  is  rotated  n 
times  per  second  about  an  axis  00',  which  bisects  the  coil  and  is 
perpendicular  to  the  field  (Fig.  382  a,  6).  Let  l  =  AB  =  DC,  and 
r  =  AO  =  DO.  The  direction  of  rotation  looking  on  the  end  AOD  is 
anti-clockwise.  Let  XOX'  and  YOY'  represent  the  planes 
through  0,  respectively  parallel  and  at  right  angles  to  the  mag- 
netic field.  The  side  A  B  is  evidently  cutting  lines  of  force  most 
rapidly  at  X  and  X',  where  it  is  moving  at  right  angles  to  the 
lines;  while  at  Y  and  Yf  it  is  moving  parallel  to  the  field,  so  it 
is  cutting  no  lines,  and  hence  has  no  e.m.f.  induced  in  it  at  this 
instant.  The  e.m.f.  induced  in  A  B  is  a  maximum  at  X,  and 


ELECTROMAGNETIC  INDUCTION 


451 


zero  at  Y,  a  negative  maximum  at  X',  and  zero  again  at  Y'. 
Thus  the  e.m.f.  in  A B  is  from  A  to  B  while  the  coil  is  mov- 
ing from  Y  through  X  to  y ;  it  is  from  B  to  A  while  moving 
from  Y'  through  X'  to  Y.  (These  directions  follow  in  accord- 
ance with  Fleming's  rule,  §502.)  The  e.m.f.  induced  in  the 
opposite  side  DC  evidently  adds  itself  to  that  in  A  B  to  pro- 
duce a  single  alternating  current  in  the  circuit  A  BCD.  The 
ends  BC  and  DA  do  not  cut  lines,  and  hence  have  no  e.m.f. 
induced  in  them.  There  is  thus  induced  in  the  coil  A  BCD 
an  e.m.f.  which  goes  through  a  complete  cycle  once  in  a  revolution. 
It  can  be  shown  that  this  e.m.f.  at  any  instant  is  proportional 
to  the  sine  of  the  angle  6  which  the  plane  of  the  coil  makes  at 


-„      (*) 


Fid.  382 a. 


Fia.  3826. 


that  instant  with  the  plane  perpendicular  to  the  field.  Thus  in 
Fig.  3826,  the  e.m.f.  is  equal  to  e  =  E  sin  6,  where  E  is  the  maxi- 
mum e.m.f.  induced,  and  6  is  the  angle  of  the  coil  with  the  plane 
YOY'. 

514.  Formula  for  the  E.  M.  F.  in  a  Rotating  Coil.— Let  V  = 
the  uniform  tangential  speed  of  AB  (and  CD).  At  the  instant 
when  the  angle  between  the  coil  and  the  plane  perpendicular  to 
the  field  is  0,  this  velocity  is  represented  by  AR,  Fig.  383.  The 
velocity  component  at  right  angles  to  the  field  is  RS  =V  sin  6. 
Let  #  =  the  strength  of  the  field  (  =  the  number  of  magnetic  lines 
per  square  centimeter) ;  then  VIH  sin  6  is  the  number  of  magnetic 
lines  cut  by  AB  (—1)  per  second.  Hence  the  e.m.f.  induced 
in  AB  and  CD  is  e  =  2VlH  sin  6  =  E  sin  6.  Here  E  =  2VIH  = 
e.m.f.  when  the  coil  is  passing  through  the  points  X  and  X' 
where  it  is  cutting  the  lines  at  the  maximum  rate,  or  when  6  —  90° 
or  =270°.  The  curve,  Fig.  384,  represents  this  e.m.f.  during  a 
single  rotation.  The  ordinates  are  proportional  to  the  e.m.f. 


452 


ELECTRICITY  AND  MAGNETISM 


and  the  abscissas  are  proportional  to  the  angles.  Since  the 
rotation  is  uniform,  the  abscissas  are  also  proportional  to  the 
time.  Ordinates  above  the  line  represent  electromotive  forces 
in  one  direction,  and  ordinates  below  represent  electromotive 
forces  in  the  reverse  direction.  That  is,  a  rectangular  coil  rotated 
uniformly  in  a  uniform  magnetic  field,  has  induced  in  it  an  alter- 
nating e.m.f.  which  varies  as  the  sine  of  an  angle,  and  is  repre- 
sented by  a  sinusoidal  curve. 


\ 


9V 


FIG.  383. 


\ 


Fio.  384. 


The  formula  e  =  2VlH  sin  d,  can  be  changed  into  the  form 
e  =  2nn  NQ  sin  0,  where  n  is  the  number  of  revolutions  per  second, 
and  Af0  is  the  total  magnetic  lines  through  the  coil  when  it  is  at 
right  angles  to  the  field.  To  prove  this,  put  V  =  2nnr,  and  we 
get  e  =  ±nnrl  H  sin  d.  But  2rZ  =  S,  the  face  area  of  the  coil,  and 
SH  =  N0.  Hence  e  —  'lnn  N0  sin  6.  It  is  easy  to  show  that  this 
equation  holds  for  any  shape  of  the  rotating  coil. 

The  varying  e.m.f.  induced  in  a  rotating  coil  of  any  form  and  of  area  S 
can  be  obtained  from  the  relation  E  -  -  -^.  N  is  the  total  number  of 

lines,  or  Rux,  of  magnetic  force  through  the 
coil  and  is  evidently  equal  to  the  QUK  through 
the  projection  of  the  coil  on  the  plane  YY'  or 

HS  cos  e.  Hence  E  -  HS  sin  $  -^  -  HSusin  e 
where  w  ••  the  angular  velocity  =•  2*-n. 

515.  The  Earth  Inductor. — The  earth 
inductor  is  a  coil,  Fig.  385,  usually  of 
a  large  number  of  windings,  which  is 
mounted  so  that  it  can  be  rotated  about 
either  a  horizontal  or  a  vertical  axis. 
Suppose  the  axis  vertical  and  the  plane 
of  the  coil  at  right  angles  to  the  magnetic  meridian.  By  revolv- 


Fia.  385. 


ELECTROMAGNETIC  INDUCTION 


453 


ing  the  coil  through  180°,  the  magnetic  flux  is  taken  out  of  one 
face  and  put  in  at  the  other  face.  Let  H=  the  horizontal  in- 
tensity of  the  earth's  magnetic  field,  S  =  the  area  of  the  coil 
face,  and  R  =  the  resistance  of  the  circuit.  Then  the  quantity 
of  electricity  flowing  in  the  coil  during  a  rotation  of  180°,  is 
q=(2HS)/R  (§505).  This  can  be  measured  by  the  throw  of  a 
ballistic  galvanometer.  Similarly  when  the  axis  is  placed  hori- 
zontally, q'  =  (2VS)/R.  where  V  is  the  vertical  component  of 
the  earth's  field.  We  thus  get  (q'/q)  =  7/#  =  tan  0,  where  0 
is  the  dip  or  inclination  (§386). 

516.  Use  of  Induced  Currents  to  Compare  Fields  and  to  Measure  Magnetic 
Induction. — From  the  above  we  see  that  the  intensities  of  two  magnetic 
fields  can  be  compared  by 

quickly  rotating  the   same  Bt 

coil  in  the  two  fields  and  ^ 1|-|| 

comparing  the  throws  of  a 
ballistic  galvanometer. 
More  often  a  small  coil, 
often  termed  a  "flip  coil"  is 
quickly  jerked  out  of  the 
field,  and  the  throw  of  the 
galvanometer  noted. 

The  change  of  magnetic 
induction  B,  in  a  ring  mag- 
net is  determined  in  Row- 
land's method  of  obtaining 
BH  curves,  by  observing  the 
galvanometer  throw  due  to  the  current  induced  in  a  secondary  coil  Sc.  The 
arrangement  is  shown  in  Fig.  386.  The  iron  anchor  ring  is  wound  uniformly 
with  the  magnetizing  coil  Pr.  The  current  is  measured  by  the  ammeter 
Am,  and  regulated  by  the  rheostat  R.  Changes  in  the  current  in  Pr  produce 
changes  in  the  magnetic  induction  B  through  the  iron.  Starting  from  zero 
the  current  is  increased  by  steps,  and  the  increments  in  the  induction  B  are 
calculated  from  the  galvanometer  throws.  The  total  induction  is  gotten  by 
adding  the  increments.  The  magnetizing  force  H  is  calculated  from  the 
dimensions  of  the  coil  Pr  and  the  current  f.  The  ballistic  galvanometer  G 
is  calibrated  by  observing  the  throw  from  revolving  the  earth  inductor  7  in 
the  earth's  magnetic  field.  The  values  of  H  and  the  corresponding  values 
of  B  thus  determined  are  then  plotted  to  give  the  usual  curves  (§490). 

517.  Simple  Alternating  Current  Dynamo. — In  Fig.  387,  we 
have  a  coil  revolving  in  the  field  between  the  poles  of  a  magnet. 
The  ends  of  the  coil  are  connected  with  the  insulated  metal  rings 


Fio.  386. 


454 


ELECTRICITY  AND  MAGNETISM 


N  and  M,  on  the  shaft  00'.  Metal  springs  or  "brushes"  rest  or 
slide  on  these  collector  rings.  Thus  the  current  induced  in  the 
coil  A  BCD  flows  through  the  external  circuit.  Such  a  machine 
forms  a  simple  alternating  current  dynamo.  By  winding  the 
coil  on  an  iron  cylinder,  the  intensity  of  the  magnetic  field  is 
increased,  and  thus  a  greater  e.m.f.  is  induced.  But  this  iron 
core  being  itself  a  conductor  will  have  eddy  currents  induced  in 
it,  unless  it  is  laminated,  so  as  to  make  the  resistance  infinitely 
great  in  the  direction  of  the  induced  e.m.f.  Hence  the  core  is 
built  up  of  insulated  iron  disks  as  represented  in  Fig.  388. 
Commercial  alternating  current  dynamos  are  always  multipolar. 
One  of  the  reasons  for  this  is  that  for  practical  electric  lighting 
and  power  transmission,  frequencies  of  from  25  to  125  alterna- 


Fia.  387. 


Fia.  388. 


Fio.  389. 


tions  per  second  are  desirable.  A  common  frequency  for  most 
purposes  is  now  60  alternations  per  second.  To  get  such  fre- 
quencies with  safe  speeds  it  is  necessary  to  have  field  magnets 
with  multiple  poles.  Fig.  389  shows  a  multipolar  alternating 
current  dynamo. 

618.  Simple  Direct  Current  Dynamo. — To  obtain  a  current  con- 
stant in  direction,  a  commutator  is  used  instead  of  collector  rings. 
Fig.  390  represents  a  two-part  commutator.  This  consists  of  a 
copper  ring,  which  has  been  cut  into  half  rings.  These  half  rings 
are  insulated  and  form  the  ends  of  the  coil  A  BCD.  The  brushes 
R  and  S  are  set  180°  apart,  so  that  one  rests  on  one-half  of  the 
commutator,  while  the  other  rests  on  the  opposite  half.  These 
brushes  thus  make  the  connections  for  the  current  with  the 
external  circuit.  The  brushes  are  set  so  that  the  connections 
with  the  external  circuit  are  reversed,  just  at  the  instant  in  which 


ELECTROMAGNETIC  INDUCTION 


455 


the  current  in  the  rotating  coil  is  reversed  (that  is,  approximately 
as  the  coil  passes  through  the  plane  perpendicular  to  the  field). 
The  current  in  the  external  circuit  is  thus  uni-directional,  and 
varies  as  represented  in  Fig.  391.  The  above  is  a  simple  direct 
current  (D.  C.)  dynamo  with  a  two-part  commutator.  In  Fig. 
392  we  have  two  coils  at  right  angles  to  each  other;  by  joining  to 
a  proper  commutator  we  obtain  in  the  external  circuit,  the 
current  represented  in  Fig.  393.  This  is  the  sum  of  two  pulsating 


\ 


\/ 


\ 


Fio.  390. 


Fio.  391. 


currents,  which  differ  in  phase  by  90°.  It  is  seen  that  the  per 
cent,  of  variation  is  much  less  than  in  the  current  from  a  dynamo 
with  a  two-part  commutator.  In  modern  direct  or  continuous 
current  dynamos  there  are  often  hundreds  of  coils,  joined  to  a 
commutator  of  many  sections,  and  the  resulting  current  is  prac- 
tically constant.  The  inductance  of  the  circuit  also  operates  to 
lessen  the  variations  of  the  current  in  these  machines. 


Fio.  392. 


Fio.  393. 


519.  Dynamo -electric  Machines. — The  parts  of  a  dynamo- 
electric  machine,  or  a  dynamo,  are  (1)  the  field  magnets  for 
producing  the  magnetic  field;  (2)  the  armature,  or  the  coils  in 
which  the  currents  are  induced.  The  armature  coils  are  nearly 
always  on  a  laminated  iron  core,  and  are  supplied  with  slip  rings 
or  a  commutator,  and  brushes  to  make  connection  with  the 
external  circuit.  In  the  simple  forms  of  dynamos  described  in 


456 


ELECTRICITY  AND  MAGNETISM 


previous  sections,  the  armature  revolves,  and  the  field  magnets 
are  stationary.  In  large  dynamos  for  high  electromotive  forces, 
the  armature  is  often  made  the  stationary  part  and  the  field 
magnets  revolve.  In  one  type  of  A.  C.  dynamos,  the  revolving 


FIG.  304. 


parts  are  iron  masses  which  change  the  magnetic  flux  through 
the  armature  coils,  the  magnet  coils  being  also  stationary.  This 
type  of  dynamo  is  called  the  "inductor"  form. 

Dynamos  are  "direct"  current  (D.  C.),  or  alternating  current 
(A.  G.),  according  to  the  character  of  the  e.m.f.  at  the  terminals 


Fio.  395. 

of  the  machine.  The  field  magnets  may  be  bi-polar  or  multi- 
polar.  Several  common  forms  of  field  magnets  are  shown  in 
Fig.  394.  In  some  of  the  early  dynamos,  permanent  magnets 
were  used  for  field  magnets.  Such  machines  were  called  mag- 
neto-electric machines  or 
magnetos.  The  small  ma- 
chines often  used  in  telephone 
call  boxes  are  magnetos. 
But  the  field  magnets  of  all 
modern  power  dynamos  are 
electromagnets. 

Fig.  395,  a,  6,  c,  d,  shows  the 

Fio    396. 

different  methods  in  which  the 

field  magnets  are  excited.  These  are  (a)  separately  excited, 
that  is  the  current  in  the  field  coils  comes  from  a  separate 
generator;  (6)  "series  wound,"  that  is  the  field  coils  are  in  series 


ELECTROMAGNETIC  INDUCTION  457 

with  armature  and  the  external  circuit,  so  that  all  the  current  of 
the  armature  passes  through  the  field  coils  as  well  as  the  external 
circuit;  (c)  "shunt  wound,"  that  is,  the  field  coils  and  the  ex- 
ternal circuit  are  in  parallel,  so  that  only  a  part  of  the  current  of 
the  armature  passes  through  the  field  coils;  (d)  "compound 
wound,"  that  is,  part  of  the  field  coils  are  series  and  part  shunt 
windings.  The  choice  of  windings  of  the  field  coils  is  largely  a 
question  of  regulation  of  the  electromotive  force  under  different 
loads.  For  a  discussion  of  these  methods,  the  student  must 
consult  special  manuals. 

The  two  most  common  forms  of  D.  C.  armatures  are:  (a)  the 
ring  armature,  sometimes  called  the  Gramme 
armature,  after  its  inventor,  and  (6)  the  drum 
armature.     The  ring  armature  is  represented 
diagrammatically  in  Fig.  396.     The  coils  are 
wound  around  a  closed  ring  of  soft  iron,  and 
connected  as  indicated.     The  core  of  the  ring 
is  laminated.     Only  the  wires  on  the  outside  of  the  ring  are 
inductors,  as  the  wires  on  the  inside  are  shielded  magnetically. 
The  course  of  the  lines  of  force  is  indicated  in  the  figure. 

The  first  and  the  simplest  form  of  drum  armature  is  the 
"shuttle"  armature  used  by  Siemens  in 
his  machine  of  1856.  A  section  is  shown 
in  Fig.  397.  The  iron  core  increases  the 
magnetic  flux  through  the  coil.  This 

*orm  *s  now  usec*  on*y  *n  sma^  magnetos. 
The  winding  and  commutator  connec- 
tions of  a  modern  drum  armature  are  very  complicated.  The 
coils  may  be  on  the  surface  or  in  tunnels  or  grooves  in  the 
core  of  the  armature.  The  magnetic  lines  go  from  pole  to  pole 
as  indicated  in  Fig.  398.  For  the  study  of.  the  forms  of  armature 
and  of  their  actions  and  reactions,  the  student  must  consult 
special  treatises  on  dynamo-electric  machinery. 

520.  The  Alternating  Current  Transformer. — The  alternating 
current  transformer  is  a  form  of  induction  coil,  used  for  trans- 
forming alternating  currents  of  one  potential  into  alternating 
currents  of  a  different  potential.  It  consists  of  a  primary  coil 
Pr,  and  a  secondary  coil  Sc,  and  a  laminated  iron  core  to  increase 
the  magnetic  flux.  It  is  most  commonly  used  to  "step  down" 


458  ELECTRICITY  AND  MAGNETISM 

from  a  higher  voltage  to  a  lower  voltage.  The  energy  of  the 
secondary  current  in  well-designed  transformers,  is  equal  within 
a  small  percentage  to  the  energy  of  the  primary  circuit.  Thus  a 
current  of  one  ampere  at  1000. volts  is  transformed  into  approxi- 
mately 10  amperes  at  100  volts.  The  Pr  coil  has  in  this  case  ten 
times  the  number  of  turns  of  the  Sc  coil.  The  only  limit  to  the 
potentials  that  can  be  obtained  with  transformers  is  that  of  in- 
sulation. The  coils  of  tansformers  for  high  potentials  are  gen- 
erally immersed  in  a  high  insulating  mineral  oil. 

621.  Advantages  of  Alternating  Currents  in  Power  Transmis- 
sion.— Within  recent  years,  electric  power  has  been  transmitted 
scores  of  miles,  and  alternating  currents  are  used  exclusively  on 
these  long  distance  power  lines.  The  reasons  for  this  general  use 
of  the  alternating  current  in  transmitting  electric  energy  over 
longer  distances  are  (a)  the  ease  of  transforming  the  alternating 
current  from  high  to  low  potentials;  (6)  the  possibility  of  securing 
high  insulation  in  alternating  current  machinery;  (c)  the  inven- 
tion of  the  A.  C.  induction  motor  (§538). 

Electric  power  is  measured  by  the  product  of  the  current  and 
of  the  potential,  or  equals  le;  thus  the  same  power  can  be  trans- 
mitted at  a  high  potential  with  a  small  current,  or  at  a  low  poten- 
tial with  a  correspondingly  larger  current.  But  the  weight  of 
copper  needed  in  the  lines  increases  rapidly  with  the  current,  since 
heating  effects  vary  as  PR  (§458).  There  is  thus  a  great  econ- 
omy in  the  transmission  of  electric  power  at  high  potentials,  and 
indeed  only  in  this  way  is  it  commercially  possible.  Potentials 
of  from  30,000  to  60,000  volts  are  in  use  for  such  transmission. 
But  these  high  potentials  cannot  be  used  safely  in  lamps  or  in 
moving  apparatus,  so  that  it  is  necessary  to  transform  to  lower 
potentials  before  using  the  currents.  For  alternating  currents, 
this  can  be  easily  and  efficiently  done  by  the  A.  C.  transformer 
(§520).  Such  changes  of  potentials  are  not  possible  with  any 
D.  C.  apparatus.  The  use  of  high  potentials  with  alternating 
currents  is  also  possible,  because  A.  C.  machinery  can  be  insulated 
to  stand  the  highest  potentials.  The  A.  C.  dynamo  has  no 
commutator  and  the  armature  can  in  addition  be  made  the 
stationary  part.  The  A.  C.  induction  motor  (§538)  also  shares 
in  the  advantages  of  high  insulation  as  well  as  efficiency  and 
simplicity. 


ELECTROMAGNETIC  INDUCTION 


459 


Fio.  399. 


!.  Two    Phase    and  Three   Phase     A.   C.  Dynamos. — The 

alternating  current  dynamo  is  often  made  so  as  to  generate  more 
than  one  alternating  current.  The  alternating  currents  in  such 
machines  always  differ  in  phase,  and  so  they  are  known  as  polyphase 
currents.  The  only  systems  in  commercial  use  are  the  two  and 
three  phase  systems.  The  currents  are  generated  in  coils  placed 
at  different  angles  on  the  armature.  Thus  in  a  two  phase 
machine,  there  are  two  sets  of 
armature  coils,  the  first  set  cut- 
ting the  magnetic  field  at  a 
maximum  rate,  when  the  in- 
duced e.m.f.  in  the  second  set 
is  zero,  etc.  Fig.  399  shows  an 
arrangement  of  coils  for  a  simple 
bi-polar  machine  by  which  two 
such  currents  could  be  gener- 
ated. These  currents  differ  in 

phase  by  a  quarter  period  or  90°,  and  so  a  system  using  such 
currents  is  called  a  "quarter  phase"  system.     Fig.  400  shows 
the  phase  of  two  such  currents.     Fig.  401  represents  the  phase 
relations  of  the  three  currents,  where  the  phase  difference  is 
120°.     A   machine  wound   to   generate 
only  one  alternating  current  is  called  a 
single-phase  machine,  to  distinguish  it 
from    the    polyphase    machines.      The 
frequencies  used    with  two    and    three 
FIO.  400.  phase  alternating  currents,  are  the  same 

as  with  single  phase  alternating  currents, 

that  is,  60  alternations  per  second  for  ordinary  conditions,  with  as 
low  as  25  for  power  purposes  alone.  As  already  explained  the  com- 
mercial machines  are  always  multipolar  for  mechanical  reasons. 

The    advantage    of    two    and    three 
phase  alternating  currents  is  that  the  in- 
duction motor  can  be  used.     A  polyphase 
machine  has  also  generally  a  larger  out- 
put   for    weight    of    machine    than    a  FIO.  401. 
single  phase  machine.     The  relative  advantage  of  the  two  and 
three  systems,  lies  in  the  size  of  conductors  required  for  the  dis- 


X 

-^ 

^ 

\ 

s~- 

—  -s 

c/ 

\ 

s*~ 

~t> 

\ 

/ 

\ 

/ 

\ 

7 

^< 

/ 

\ 

^^ 

^s 

/ 

\ 

^. 

—^ 

/ 

v 

460 


ELECTRICITY  AND  MAGNETISM 


tribution  of   a  given  power.     For  a  discussion  of  this,  special 
treatises  must  be  consulted. 

523.  Effect  of  Inductance  in  an  A.  C.  Circuit. — When  an  alter- 
nating e.m.f.  acts  in  a  non-inductive  circuit,  the  current  /  and 
the  e.m.f.  are  in  the  same  phase  as  is  represented  in  Fig.  402. 
Here  the  current  /  at  any  instant  is  equal  to  E/R,  where  R  is 
resistance  of  the  circuit,  and  E  is  the  e.m.f.  in  the  circuit  at  the 
instant.  When  the  e.m.f.  is  zero,  the  current  is  zero,  and  the 
maximum  current  corresponds  in  time  to  the  maximum  e.m.f., 
etc.  If  the  circuit  is  inductive,  then  experiments  with  the  oscillo- 


fk 


41 

r  — 

IR 

l3 

x' 

\ 

s 

1  

V  / 

"v 

X, 

x 

s\ 

/ 

f 

s>. 

v^ 

\ 

\ 

- 

^ 

// 

"\ 

~> 

•'' 

s' 

Fio.  402. 


Fio.  403. 


graph  (§524)  show  that  the  current  lags  behind  the  external  or 
impressed  e.m.f.,  that  is,  the  current  reaches  a  maximum  later 
than  the  impressed  e.m.f.  This  lag  is  due  to  self-induction.  We 
have  seen  that  starting  or  increasing  a  current  in  an  inductive 
circuit  produces  an  e.m.f.  of  self-induction  which  tends  to  retard 
the  current  growth,  and  similarly  breaking  or  decreasing  a  cur- 
rent produces  an  e.m.f.  of  self-induction  which  tends  to  prolong 
the  current.  Thus  the  actual  e.m.f.  at 
any  instant  in  an  inductive  circuit  is  the 
algebraic  sum  of  the  external  or  im- 
pressed e.m.f.  (produced  by  generator, 
etc.),  and  the  e.m.f.  of  self-induction. 
This  e.m.f.  of  self-induction  is  equal  to 
-L(dl/d£)  (§509). 

In  Fig.  403  we  have  the  curves  of  the 
impressed  e.m.f.  E,  of  the  effective  e.m.f.,  which  is  equal  to  IR, 
and  of  the  e.m.f.  of  self-induction  —  Ldl/dt,  represented  in  their 
phase  relations. 

In  the  coil  of  an  electromagnet,  where  the  inductance  is  large,  the 
e.m.f.  of  self-induction  is  correspondingly  large,  and  this  maybe  sufficient  to 
make  the  effective  e.m.f .  practically  zero.  Such  a  coil  is  called  a  choking  coil, 


Fio.  404. 


ELECTROMAGNETIC  INDUCTION 


461 


or  an  impedance  coil.  The  resistance  of  an  inductive  circuit  to  an  alternating 
current  is  called  impedance.  It  can  be  shown  that  the  impedance  of  a 
circuit  is  equal  to  \//23  +  4^3n2La,  where  R  is  the  resistance,  L  is  the  in- 
ductance of  the  circuit,  and  n  is  the  frequency  of  the  alternating  current. 
The  proof  of  this  is  given  in  treatises  on  alternating  currents.  This  fact 
of  impedance  explains  why  little  current  goes  through  the  primary  of  a 
transformer  when  the  secondary  circuit  is  not  closed. 

It  is  to  be  noticed  that  the  impedance  increases  with  the  frequency 
n.  The  frequency  of  a  Leyden  jar  discharge  is  ordinarily  very  high 
(§  541),  and  so  a  discharge  has  a  large  impedance  even  through  a  single 
loop.  Thus  in  Fig.  404  the  discharge  will  leap  across  a  considerable  air- 
gap,  sooner  than  go  through  the  loop  of  wire. 

524.  The  Oscillograph. — An  ordinary  galvanometer  shows  no 
deflection  from  an  alternating  current,  because  the  needle  system 
has  so  much  inertia  that  it  cannot  follow  the  rapid  impulse  from 
the  alternating  current.  Blondel,  Duddell,  and  others  have 
made  galvanometers  with  very  light  moving  parts  and  of  high 
frequency,  so  that  the  needle 
system  can  follow  the  changes  in 
the  alternating  currents.  A  gal- 
vanometer with  a  high  frequency 
needle  system,  so  that  its  de- 
flections show  the  variations  in 
alternating  currents,  is  an  oscillo- 
graph. Fig.  405  shows  diagram- 
matically  one  of  the  best  forms  of 
oscillographs.  It  consists  of  a 
narrow  loop  of  phosphor  bronze 
strip,  which  is  stretched  with 

considerable  tension,  by  a  spring  s,  so  as  to  have  a  very  short 
natural  period  of  vibration.  This  is  in  the  strong  magnetic  field 
and  placed  with  the  plane  of  the  loop  parallel  to  the  magnetic 
field.  The  strip  is  thus  twisted  by  the  magnetic  forces  when  a 
current  passes  through  the  loop.  The  natural  frequency  of  the 
loop  is  commonly  from  3 ,000  to  10,000  vibrations  per  second,  so  that 
its  deflections  can  follow  very  closely  all  ordinary  variations  in 
alternating  currents.  The  deflections  are  recorded  by  a  beam  of 
light  which  is  reflected  from  a  small  mirror  M,  attached  to  the 
loop.  This  beam  of  light  falls  on  a  photographic  plate  which 


Fio.  405. 


462  ELECTRICITY  AND  MAGNETISM 

moves  at  right  angles  to  the  deflections.     It  thus  leaves  a  curve 
showing  the  variations  of  the  current. 

Another  form  of  oscillograph  is  the  Braun  cathode  tube.  The  current 
passes  through  a  helix,  and  thus  acts  magnetically  on  a  pencil  of  cathode 
rays  in  a  special  form  of  Crookes  tube  (Fig.  406).  When  not  deflected, 
this  pencil  of  cathode  rays  produces  a  luminous  spot  on  a  phosphorescent 
screen  in  one  end  of  the  cathode  tube.  Under  the  action  of  an  alternating 
current  through  the  helix  the  luminous  spot  from  the  cathode  rays  vibrates 
in  a  luminous  line  on  the  phosphorescent  screen.  Looked  at  in  a  rotating 
mirror  this  is  drawn  out  into  an  alternating  current  curve.  The  same 
may  be  photographed  on  a  plate  moving  at  right  angles  to  the  vibrations  of 
the  luminous  spot. 


Fio.  406.  FIG.  407. 

525.  The  Telephone. — The  telephone,  invented  by  Bell  in  1876, 
consists  of  a  thin  iron  plate  or  membrane,  supported  in  front  of 
the  pole  of  a  permanent  magnet,  and  a  spool  of  wire  over  the 
magnet  pole  (Fig.  407).     Sounds  can  be  transmitted  electrically 
to  a  distance  by  using  two  telephones,  one  for  a  transmitter, 
and  the  other  for  the  receiver.     The  two  wire  spools  are  con- 
nected in  series  by  the  wires  joining  the  two  stations.     The  sound 
waves  set  the  thin  iron  plate  in  vibration,  and  the  approach  or 
receding  of  this  plate  changes  the  magnetic  flux  through  the  coil. 
This  induces  currents  in  the  coils  and  the  line  which  undulate  in 
unison  with  and  in  proportion  to  the  sound  waves.     These  cur- 
rents strengthen  and  weaken  the  attraction  of  the  magnet  of  the 
receiver,  and  thus  produce  vibrations  of  the  receiver  plate  which 
correspond  to  the  vibrations  of  the  transmitter  plate.     The 
electric  currents  induced  in  the  above  cases  are  very  feeble,  and 
can  transmit  sounds  only  short  distances.     For  longer  distances, 
the   microphone   described   in   the   next   section  is  used  as  a 
transmitter. 

526.  The  Microphone. — The  microphone  depends  upon  a  fact 
discovered  by  Hughes  in  1878,  that  the  electrical  resistance  of  a 
loose  contact  between  two  conductors  changes  under  the  action 


ELECTRODYNAMICS 


463 


of  sound  waves.  Variations  of  the  current  can  thus  be  produced 
in  a  circuit,  these  variations  corresponding  to  the  sound  waves 
which  produce  them.  A  simple  form  of  microphone  consists  of  a 
piece  of  carbon  resting  on  two  pieces  of  carbon,  and  thus  com- 
pleting a  circuit  which  includes  a  battery  and  a  Bell  receiver. 


Fio.  408. 


Fio.  409. 


The  carbons  can  be  mounted  on  a  sounding  box.  Such  an  ar- 
rangement makes  an  effective  transmitter  (Fig.  408).  In  the 
Running's  transmitter,  which  has  been  extensively  used  in  long 
distance  telephony,  granulated  carbon  is  placed  in  between  two 
metal  plates  as  shown  in  Fig.  409. 


ELECTRODYNAMICS 

527.  Motion  of  a  Circular  Circuit  in  a  Magnetic  Field.  Max- 
well's Rule. — If  a  conductor  carrying  a  current  is  placed  in  a 
magnetic  field  (§  368),  there  is  in  general  a  force  tending  to  move 
the  conductor.  It  has  been  shown 
(§427)  that  a  circular  circuit  or 
other  plane  closed  circuit,  acts  like 
a  magnet,  and  that  the  lines  offeree 
due  to  the  current  enter  the  face 
which  the  current  flows  clock- 
,  and  emerge  from  the  face  in 


in 
wise 


FIG.  410. 


which  the  current  flows  anti-clock- 
wise (Fig.  410).  When  such  a  coil  is  placed  between  the  poles 
of  a  magnet,  the  coil  tends  to  place  itself  so  that  its  plane  is 
at  right  angles  to  the  field,  the  clockwise  face  of  the  coil  being 
toward  the  N  pole;  in  other  words,  the  coil  places  itself  so  that 
the  lines  of  force  of  the  field  and  of  the  coil  coincide.  Maxwell 
has  generalized  this  into  a  rule — An  electric  circuit  tends  to 


464 


ELECTRICITY  AND  MAGNETISM 


FIG.  411. 


move  in  a  magnetic  field  so  as  to  include  the  maximum  number  of 

lines  of  force.  Thus  the  lines  of 
the  field  and  of  the  circuit  are  in 
the  same  direction  when  stable 
equilibrium  is  reached 

628.  Force  on  a  Linear  Circuit 
in  a  Magnetic  Field.— In  Fig.  411 
is  represented  an  experiment  for 
showing  the  action  between  a 
linear  circuit  and  a  magnetic  field 
at  right  angles  to  the  circuit. 
A  B  is  a  strip  of  flexible  copper 
foil  hanging  vertically  and  mak- 
ing connection  with  the  mercury 
cup  C.  The  current  flows  from  A  to  B  as  indicated.  The  mag- 
netic field  is  that  of  a  U-shaped  magnet  and  is  horizontal.  The 
flexible  conductor  is  acted  on  by  a  force  at  Motion 
right  angles  to  the  plane  of  the  field  and  of 
the  current.  The  relative  directions  of  the 
current,  the  field  and  the  motion,  are  shown 
in  Fig.  412.  Fleming  has  given  the  following 
convenient  rule  for  remembering  these  direc- 
tions, the  rule  being  similar  to  that  for  in- 
duced currents  (§502).  Hold  the  left  hand 
with  the  thumb,  the  forefinger  and  the  middle  or  center  finger, 
so  that  each  is  at  right  angles  to  each  of  the  others;  then,  if  the 
forefinger  is  in  the  direction  of  the  field  and  the  center  finger  in 

the  direction  of  the  cur- 

.*     -  G  rent,  the  thumb  will  in- 

dicate the  direction  of 
the  resulting  force  on 
the  linear  circuit. 

Fig.  413  shows  the 
distribution  of  the  mag- 
netic lines  about  a  cur- 
rent which  is  at  right 
angles  to  a  uniform  magnetic  field  (A).  The  current  flows  down, 
at  right  angles  to  the  plane  of  the  figure,  and  hence  its  magnetic 
lines  are  clockwise  (B).  This  is  compounded  with  the  uniform 


FIG.  412. 


FIG.  413. 


ELECTRODYNAMICS 


465 


magnetic  field,  strengthening  the  field  on  the  side  C  and  weak- 
ening it  on  the  side  D.    The  movement  of  the  circuit  in  the 
direction  C  to  D,  can  be  considered  as  due  to  the  tendency  of 
the  lines  on  the  side  C  to  contract.    The 
addition  of  the  fields  is  indicated  at  P. 

529.  Numerical  Value  of  the  Force  on  a  Linear 
Circuit  in  a  Magnetic  Field. — We  have  seen  in 
§428  that  a  circuit  element  ids  acts  on  a  magnetic 
pole  m  at  a  perpendicular  distance  r  with  a  force 

F  = — r — .  There  is  evidently  an  equal  and  op- 
positely directed  force  acting  on  ids  due  to  the 
field  on  m,  that  is  a  force  —F,  which  is  at  right 
angles  to  the  plane  of  the  pole  and  the  circuit.  FIO.  414. 

The  field  due  to  m  is  H  =  m/r8.     Hence  the  force  on 

ids  is  F**Hids.  For  a  circuit  of  length  L,  F=*HiL.  That  is,  a  linear  cir- 
cuit Li  at  right  angles  to  a  uniform  magnetic  field  H  is  acted  on  by  a  force  of 
Hil  dynes,  and  this  force  is  at  right  angles  to  the  plane  of  H  and  iL.  If 
iL  is  not  at'  right  angles  to  H,  iL  sin  6  the  component  of  iL  at  right  angles  to 
H  is  to  be  taken,  where  6  is  the  angle  between  H  and  iL. 

630.  Force  Between  Two  Parallel  Circuits. — If  two  circular 
coils  carrying  currents  are  hung  by  flexible  wires  parallel  to  each 
other  (Fig.  414)  they  attract  each  other  when  the  currents  in  the 
two  coils  are  in  the  same  direction,  and  repel  each  other,  when  the 
currents  are  in  opposite  directions.  It  is  easy  to  see  that  each  of 
the  circuits  thus  tends  to  move  so  as  to  include  the  maximum 
number  of  lines  of  force.  Rectangular  or 
other  plane  closed  circuits  can  be  substituted 
for  the  circular  circuits. 

Lord  Kelvin  makes  use  of  the  forces  between 
parallel  circuits  in  his  electrodynamic  balance 
for  measuring  electric  currents. 

631.  Force  Between  Two  Circular  Currents 
at  Right  Angles. — Two  circular  currents  at 
right  angles  to  each  other,  tend  to  rotate  and  place  themselves 
in  the  same  plane,  and  with  the  currents  in  the  same  direction. 
Ampere,  who  was  the  first  to  study  the  actions  of  currents  on 
currents,  used  his  "electrodynamic  apparatus"  shown  in  Fig.  415. 
The  two  mercury  cups  a  and  b,  are  at  the  ends  of  the  conducting 


m 


FIO.  415. 


466 


ELECTRICITY  AND  MAGNETISM 


Fio.  416. 


supports  A  and  B,  and  a  is  vertically  over  b.     The  wire  frame 
mnop  is  bent  so  that  its  ends  dip  in  the  mercury  cups,  and  the 

wire  frame  is  thus  free  to  rotate  about  a 
vertical  axis  through  ab. 

The  force  of  rotation  between  coils  is  used 
in  the  electrodynamometer  for  measuring  cur- 
rents. Fig.  416  shows  the  Siemens'  form  of 
the  electrodynamometer.  The  coils  are  kept 
at  right  angles  to  each  other  by  putting  more 
or  less  torsion  in  a  spiral  spring  which  is 
attached  to  the  movable  coil.  The  amount 
of  torsion  is  read  by  a  torsion  head  and  circle 
on  top.  The  force  between  the  coils  is  pro- 
portional to  the  square  of  the  current,  and 
the  opposing  force  of  torsion  is  proportional 
to  6,  the  angle  of  torsion.  Hence  /'-A;^, 
or  7=*fc\/0,'where  A;  is  a  constant  to  be  de- 
termined for  each  instrument.  This  instrument  is  adapted  to  the  mea- 
suring of  both  direct  and  alternating  currents. 

532.  Force  between  Parallel  Linear  Circuits. — By  using  his 

electrodynamic  apparatus,  Ampere  was  able  to 

show  that  two  parallel  straight  circuits  attract 

each  other  when  the  currents  are  in  the  same 

direction.     An  arrangement  of  the  apparatus  to 

demonstrate  this,  is  shown  in  Fig.  417.     In  Fig. 

418  A   and  B  are  normal  sections   across   the 

parallel   conductors,   and  the   circular  lines   of 

force  are  shown  for  the  case  of  both  currents 

flowing  away  from  the  reader.     Evidently  the 

maximum  number  of  lines  of  force  for  each 

circuit  exists  when  A   and  B  are  close  together.     Hence  we 

should  expect   attraction  from  Maxwell's  rule.     The  repulsion 

between    oppositely    directed    currents 
follows  similarly. 

Ampere  extended  the  above  law  as  fol- 
lows :  Two  straight  conductors  which  cross 
each  other  obliquely,  attract  each  other 
when  their  currents  both  flow  toward  or 

both  away  from  the  point  of  crossing;  when  one  current  flows  away 

and  the  other  toward  the  crossing  point,  they  repel  each  other. 


FIQ.  417. 


Fio.  418. 


ELECTRODYNAMICS 


467 


Fio.  419. 


533.  Roget's   Spiral. — An  interesting   case   of   the   attraction   between 
parallel  circuits  is  that  of  Roget's  spiral  (Fig.  419).     A  spiral  brass  coil  AB 
hangs  vertically,  and  its  end  B  dips  in  a  mercury  cup  M.     When  a  current 
passes  through  AB  the  parallel  coils  attract  each  other  and  lift  B,  thus 
breaking  the  circuit;  the  attraction  ceases  and  contact 

at  B  is  again  made.     This  repeats  itself  indefinitely. 

534.  Electromagnetic  Rotation. — The   first 
apparatus  to  produce  rotation  of  a  conductor 
from  electric  currents  and  magnets  was  de- 
scribed by  Faraday  in  1821.      A  glass  tube, 
Fig.  420  is  stopped  at  both  ends  with  corks, 
and  the  lower  cork  is  covered  with  mercury, 

and  has  one  pole  of  a  magnet  NS  stuck  through  it.  A  platinum 
wire  hangs  by  a  hook  from  the  upper  cork  and  dips  in 
the  mercury  below.  When  a  current  flows  through  the 
platinum  wire  the  wire  revolves  about  the  magnetic  pole, 
and  continues  as  long  as  the  current  flows.  Reversing 
the  current  reverses  the  direction  of  the  rotation.  This 
is  evidently  a  case  of  a  conductor  moving  at  right  angles 
to  the  plane  of  the  current  and  of  the  magnetic  field 
from  the  pole  (§528).  Fig.  421,  shows  a  common  form 
of  an  electromagnetic  rotation  apparatus.  Its  action 

Fia  42°  can  be  followed  from  the  above. 

535.  Barlow's  Wheel. — This  consists  of  a  metal  disk  mounted 
to  revolve  about  a  horizontal  axis  over  a  mercury  trough  (Fig. 
422).     The  lower  edge  of  the  disk  touches  the  surface  of  the  mer- 


Fio.  422. 


cury.  A  U-shaped  magnet  lies  so  that  its  lines  of  force  cut  the 
lower  half  of  the  disk  at  right  angles.  When  an  electric  current 
flows  from  the  axis  through  the  disk  to  the  mercury,  the  disk 


468  ELECTRICITY  AND  MAGNETISM 

rotates.  Reversing  the  direction  of  either  the  current  or  of  the 
magnetic  field  reverses  the  direction  of  rotation.  A  star-pointed 
wheel  as  shown  in  the  figure  is  often  used  instead  of  the  solid  disk, 
in  order  to  reduce  the  friction  at  the  mercury  surface.  Each 
radius  of  the  wheel  in  turn  as  it  dips  in  the  mercury  becomes  a 
part  of  the  circuit,  and  is  acted  upon  by  a  force  at  right  angles  to 
plane  of  the  magnetic  field  and  the  current  (§528),  and  thus 
causes  continuous  rotation.  Barlow's  wheel  and  Faraday's  disk 
(§506),  are  evidently  inverse  machines,  the  first  using  electrical 
energy  to  produce  mechanical  motion,  and  the  second  using 
mechanical  energy  to  produce  electrical  energy. 

636.  Direct  Current  Motors. — The  direct  current  dynamo  be- 
comes a  motor  when  an  external  current  is  sent  through  its  field 
magnets  and  armature.     It  then  transforms  electrical  energy 
into  the  mechanical  energy  of  the  rotation  of  the  armature.     The 
forces  acting  on  the  armature  circuits  follow  laws  already  treated 
in  the  sections  on  the  motion  of  circuits  in  a  magnetic  field. 

The  rotation  of  the  armature  in  the  magnetic  field  induces  in 
the  armature  coils  an  e.m.f.  which  opposes  the  current  driving 
the  machine.  This  back  e.m.f.  increases  with  the  speed  of  the 
motor.  Thus  the  current  through  the  motor  decreases  as  the 
speed  increases.  When  the  speed  causes  a  back  e.m.f.  equal  to 
the  impressed  e.m.f.,  there  is  no  current  through  the  armature. 
This  occurs  when  a  frictionless  motor  runs  under  no  load,  and  is 
accordingly  doing  no  work.  At  starting,  there  is  no  motion  and 
no  back  e.m.f.,  and  hence  the  current  is  a  maximum.  To  prevent 
injury  from  the  "rush"  of  the  current  before  the  motor  reaches  a 
speed  to  produce  a  back  e.m.f.,  a  starting  resistance  is  commonly 
placed  in  series  with  the  armature.  This  starting  resistance  is 
gradually  reduced  as  the  speed  of  the  motor  increases. 

637.  Alternating  Current  Motors. — Two  similar  single-phase 
A.  C.  machines  can  be  used  as  generator  and  motor,  provided  the 
motor  is  first  brought  to  a  synchronizing  speed  with  the  genera- 
tor.    An  A.  G.  machine  thus  used  as  a  motor  is  called  a  synchron- 
ous motor.     The  synchronous  motor  is  efficient,  but  has  the  great 
disadvantage  of  not  being  self-starting,  and  also  of  stopping  if  the 
motor  is  thrown  "out  of  step"  by  overloading.     Hence  syn- 
chronous motors  are  not  in  general  use.     The  successful  use  of 
alternating  currents  for  power  transmission  has  been  due  to  the 


ELECTRODYNAMICS 


469 


Fio.  423. 


nvention  of  the  polyphase  induction  motor.  The  principle  of 
this  motor  was  first  discovered  and  stated  by  Ferraris,  in  1885, 
and  its  application  was  developed  by  Tesla  and  others.  Both 
two  and  three  phase  currents  have  been 
successfully  used  in  these  motors. 

638.  Simple  Two  Phase  Induction  Motor. — In 
Fig.  423  we  have  represented  two  sets  of  helices 
A  A'  and  BB',  placed  at  right  angles  to  each  other. 
An  alternating  current  through  the  helix  A  A' 
produces  an  alternating  magnetic  field  in  the  line 
A  A'.  If  the  current  through  AAr  is  sinusoidal, 
represented  by  the  equation  IA=!  sin  pt  (§  514) 
then  the  intensity  of  the  field  is  represented  by 

h-A  =  NI  sin  pt  =  H  sin  pt,  where  N  is  a  constant  depending  upon  the  num- 
ber of  windings  on  A  A',  etc.  Similarly,  an  A.  C.  in  BB'  will  produce  an 
alternating  magnetic  field  in  the  line  BB'.  Suppose  the  A.  C.  in  BB'  to 

differ  in  phase  from  that  in  A  A'  by  a  quarter 

phase,   that   is,  ifl=7  sin  (p«  +  90°)  =7  cos  pt. 

Then  the  field  BB'  is  hB  =>NI  cos  pt=*H  cos  pt. 
The  field  at  any  instant  is  thus  the  resultant  of 

the  two  component  fields,  hj^^H  sin  pt,  and  kg 

•=H  cos  pt.     The  resultant 

F  =  \/h  A » +  hB  2  =  \/H*  (sitfpt  +  cos'pO  =•  H 

That  is  the  resultant  is  constant  in  strength.  It  evidently  rotates  with 
every  complete  alternation  of  the  current. 

Suppose  we  place  in  this  rotating  magnetic  field,  a  "squirrel  cage"  arma- 
ture, which  can  rotate  about  its  axis  OO',  which  is  perpendicular  to  the  field. 


Fio.  424. 


Fio.  425. 


Fio.  427. 


The  construction  of  this  "squirrel  cage"  armature  can  be  seen  from  Fig.  424. 
It  consists  of  two  copper  disks  mounted  on  a  shaft  00',  and  with  copper 
bars  around  the  circumference  of  the  jiisks,  joining  the  disks.  It  is  found 


470  ELECTRICITY  AND  MAGNETISM 

that  such  an  armature  rotates  approximately  synchronously  with  the 
rotating  magnetic  field.  The  explanation  is  simple.  If  the  armature  were 
held  still,  there  would  be  electric  currents  induced  in  the  bars.  By  Lenz's 
law  these  induced  currents  would  react  magnetically  on  the  rotating  field, 
with  a  force  tending  to  prevent  the  relative  motion  of  the  armature  and  the 
field.  In  other  words,  there  is  a  torque  causing  the  armature  to  rotate  with 
the  field.  Figs.  425-7  show  a  series  of  simple  experiments  for  demonstrating 
the  principle  of  the  rotating  field,  and  the  induction  motor.  In  Fig.  425 
we  have  an  aluminum  disk,  mounted  on  a  pivot.  Under  this  is  a  magnet 
mounted  on  a  whirling  table  so  that  it  can  be  rotated.  The  disk  rotates 
with  the  magnet.  This  is  the  inverse  of  Arago's  disk  (§507),  and  the 
explanation  by  electromagnetic  induction  is  the  same  In  Fig.  426,  the 
experiment  is*varied  by  substituting  a  "squirrel  cage"  cylinder,  pivoted 
on  a  vertical  axis  between  the  poles  of  the  rotating  magnet  NS.  In  Fig.  427, 
we  substitute  for  the  rotating  magnet,  two  coils,  placed  with  the  planes  at 
right  angles.  By  passing  through  the  coils  alternating  currents,  which  differ 
in  phase  by  a  quarter  period,  a  rotating  magnetic  field  is  produced,  and  the 
"squirrel  cage"  rotates. 


ELECTRIC  OSCILLATIONS  AND  WAVES 

639.  Electric  Oscillations. — In  the  preceding  sections,  we  have 
described  alternating  electric  currents  with  frequencies  which  are 
commonly  between  25  and  125  per  second.  It  has  been  seen  that 
these  alternating  currents  follow  special  laws  which  are  due  to 
the  special  importance  of  inductance  in  such  circuits.  At  still 
higher  frequencies,  alternating  currents  bring  in  new  phenomena 
with  additional  laws.  Alternating  currents  of  high  frequency 
are  called  oscillatory  currents,  or  electric  oscillations.  The  lowest 
frequency  for  which  the  term  oscillatory  is  used  is  naturally  not 
definite,  but  we  may  in  general  think  of  an  electric  oscillation  as 
having  at  least  1,000  alternations  per  second.  It  is  often  several 
millions  per  second. 

The  study  of  electric  oscillations  has  been  in  recent  years  one 
of  the  most  important  and  fruitful  in  physics.  It  has  led  to  the 
discovery  of  electromagnetic  disturbances  in  the  space  about 
oscillatory  currents,  disturbances  which  are  propagated  outward 
as  electric  waves.  These  electric  waves  have  been  shown  to  be 
identical  physically  with  light  waves,  except  in  being  of  longer 
wave-length.  Heinrich  Hertz,  the  discoverer  of  electric  waves, 
was  thus  able  to  prove  experimentally  the  theory  of  James 


ELECTRIC  OSCILLATIONS  AND  WAVES  471 

Clerk  Maxwell,  that  light  is  an  electromagnetic  phenomenon 
(§543).  The  experiments  of  Marconi  and  others  have  resulted 
in  using  electric  waves  to  transmit  signals  by  the  electric-wave 
telegraphy.  Lodge  and  his  fellow  workers  have  also  explained 
many  of  the  "mysteries"  of  lightning  discharges  by  laws  proved 
for  oscillatory  currents. 

540.  Methods  of  Generating  Electric  Oscillations,  Alternators. — 
Two  general  methods  of  producing  high  frequency  electric  cur- 
rents or  oscillations  have  been  used,  (a)  by  multipolar  alternating 
dynamos,  and  (6)  by  an  electric  discharge  in  a  circuit  containing 
capacity  and  inductance  in  certain  ratios,  with  low  ohmic 
resistance. 

A  high  frequency  dynamo-electric  machine  must  have  a  large 
number  of  poles,  and  be  driven  at  a  high  velocity.  Such  ma- 
chines have  been  constructed  by  Tesla,  Ewing,  Duddell  and 
others.  Frequencies  of  from  10,000  to  15,000  per  second  were 
ordinarily  reached,  and  in  one  machine  a  frequency  of  120,000 
per  second  is  recorded.  But  the  velocity  of  the  moving  parts 
must  be  so  high  that  only  small  machines  are  mechanically 
possible.  The  high  frequency  alternator  has  only  recently  been 
developed  as  a  generator  of  electric  oscillations,  and  then  only 
for  special  work. 

The  generally  used  method  of  producing  oscillations  of  the 
highest  frequency  is  based  on  the  oscillatory  character  of  the  dis- 
charge of  a  circuit  containing  capacity.  This  will  be  described 
in  the  next  section. 

641.  Oscillations  by  a  Condenser  Discharge. — When  a  Leyden 
jar  is  discharged,  there  is  a  flash  which  to  the  eye  appears  as  a 
single  spark.  But  as  early  as  1842  Joseph  Henry  concluded  that 
this  discharge  of  a  Leyden  jar  "is  not  correctly  represented  by 
the  single  transfer  of  an  imponderable  fluid  from  one  side  of  the 
jar  to  the  other."  "The  phenomena,"  he  continues,  "require 
us  to  admit  the  existence  of  a  principal  discharge  in  one  direction, 
and  then  several  reflex  actions  backward  and  forward,  each 
more  feeble  than  the  preceding  until  equilibrium  is  obtained." 
Henry  reached  this  striking  conclusion  by  observing  the  irregular 
magnetization  of  steel  needles  by  Leyden  jar  discharges. 
Henry's  conclusion  was  confirmed  by  the  mathematical  theory 
of  Lord  Kelvin,  published  in  1853.  Kelvin  showed  that  the 


472  ELECTRICITY  AND  MAGNETISM 

character  of  the  discharge  depended  upon  the  resistance  R,  the 
capacity  C,  and  the  inductance  L,  and  that  the  frequency  is 
given  by  the  equation, 


R2 


If  R2I4L*  is  so  small  as  to  be  neglected  compared  to  1/LC,  then 
the  frequency  is 

1 


\ 
\ 


or  the  period  T  =  2n\/LC.     That  is,  if  the  resistance  of  the  dis- 
charging circuit  is  small,  then  the  dis- 
charge is  oscillatory.     These  oscillations 
^•^      are  rapidly  damped.     When  the  resist- 
Time  ance  R  is  large  then  the  term  under  the 

radical  is  negative  and  the  frequency  be- 
comes imaginary.  The  discharge  is  uni- 
_  Time  directional,  dying  away  slowly.  Figs. 
428,  a  and  b,  are  curves  showing  these 
two  types  of  discharge. 

In  1858  Feddersen  confirmed  Kelvin's  theory,  showing  by 
examining  the  spark  discharge  with  a  revolving  mirror  that  the 
spark  consists  of  a  series  of  alternating  and  diminishing  flashes. 
Others  have  photographed  these  flashes.  One  of  the  most 
beautiful  confirmations  of  the  oscilla- 
tory character  of  Leyden  jar  dis- 
charge is  shown  in  the  photograph 
reproduced  in  Fig.  429.  This  was 
made  by  Zenneck  in  1904,  using  a 
Braun  tube  as  an  oscillograph  (§524.) 
The  discharge  of  a  condenser  is  thus 
analogous  to  the  vibrations  of  an 
elastic  rod  clamped  at  one  end.  When 
bent  and  released,  the  rod  in  general 
vibrates  back  and  forth  about  an  FXO.  429. 

equilibrium  position,  dissipating  its  energy,  and  finally  coming 
to  rest.  But  if  the  rod  is  immersed  in  a  heavy  oil,  which  offers 
considerable  resistance  to  motion,  the  rod  comes  slowly  to  rest 
without  vibrating  beyond  its  equilibrium  position. 


ELECTRICAL  OSCILLATIONS  AND  WAVES 


473 


542.  Electric  Oscillations  and  Waves.  Resonance.— In  1888 
Heinrich  Hertz  showed  that  a  conducting  system  in  which  electric 
oscillations  are  produced  becomes  the  source  of  electric  waves, 
and  that  these  waves  can  be  detected  by  oscillations  set  up  in  a 
similar  circuit  called  a  resonator.  Fig.  430  shows  one  of  Hertz's 
arrangements.  The  discharge  rods  A  and  B  are  connected  to 
terminals  of  the  secondary  of  the  induction  coil  C,  and  are  sepa- 
rated by  the  discharge  gap  P.  The  metal  spheres  S  and  S'  slide 
on  the  rods,  so  that  the  length  of  the  discharge  circuit  can  be 
varied.  For  a  receiving  circuit  or  resonator,  Hertz  used  a  loop  of 
wire  R  broken  by  the  spark  gap  P'.  He  found  that  when  the 


P' 

•  • — 

FIG.  430. 

two  circuits  were  "  in  tune,"  a  discharge  at  P  caused  a  spark  at 
P';  or  in  other  words,  oscillations  in  the  first  circuit  produced 
oscillations  in  the  second  circuit.  The  explanation  is  evidently 
exactly  like  that  of  the  experiment  of  resonance  between  two 
tuning  forks  on  resonators.  The  sound  waves  sent  out  from  a 
tuning  fork  A  set  in  vibration  a  second  fork  B,  provided  the  two 
forks  are  of  the  same  pitch.  The  electric  waves  from  the  oscil- 
lator produce  the  electric  oscillations  in  the  resonator,  provided 
they  are  "  in  tune."  Fig.  431  shows  a  striking  class-room  experi- 
ment due  to  Lodge  for  showing  electrical  resonance.  A  and  B 
are  two  equal  Leyden  jars.  The  jar  A  has  a  wire  loop  L  which 
forms  the  discharge  circuit,  the  gap  being  between  the  polished 
balls  at  P.  The  j  ar  is  charged  by  a  small  static  electric  machine. 


474 


ELECTRICITY  AND  MAGNETISM 


The  inner  and  outer  coatings  of  the  jar  B  are  connected  by  a  wire 
loop  L' ',  the  inductance  of  which  can  be  varied  by  the  sliding  wire 
M.  By  using  a  tin-foil  strip,  a  small  gap  G  is  left  between  the 
inner  and  outer  coatings  of  B.  When  the  two  circuits  are  in  tune, 
a  discharge  in  A  produces  oscillations  in  B,  which  are  shown  by 
a  bright  spark  at  G. 


Fia.  431. 

543.  Electromagnetic  Theory  of  Light. — Using  his  spark  gap 
detector,  Hertz  showed  that  electric  waves  are  reflected  from 
plane  and  curved  metal  surfaces  in  accordance  with  the  same 
laws  as  light  waves;  that  they  are  refracted  in  passing  through 
prisms  of  resin,  paraffine  and  other  dielectrics;  that  they  are 
polarized  by  a  coarse  metal  grating,  and  hence  are  transverse 
waves.  He  measured  their  wave-length  and  computed  from  his 
oscillator  their  frequency;  and  thus,  from  the  formula  v  =  nA,  he 
determined  that  the  velocity  of  electric  waves  is  the  same  as  that 
of  light.  The  electric  waves,  which  Hertz  produced,  generally 
had  wave-lengths  of  eight  or  nine  meters.  The  shortest  electric 
wave  yet  produced  has  a  wave-length  of  about  four  millimeters, 
still  many  times  the  length  of  the  longest  infra-red  line  (§733). 

Twenty  years  before  Hertz's  experiments  were  performed, 
Maxwell  advanced  the  view  that  waves  of  light  are  electro- 
magnetic waves  of  very  short  wave-length.  From  theoretical 
calculations  Maxwell  found  that  the  velocity  of  such  waves  equal 
l/\/kjj.,  where  k  is  the  dielectric  constant  of  the  medium  and  fi 
its  permeability,  both  being  expressed  in  electromagnetic  units. 
The  velocity  thus  calculated  for  air  agrees  with  the  velocity  of 
light  (§644) .  The  value  of  p.  for  transparent  substances  is  nearly  1 . 


ELECTRICAL  OSCILLATIONS  AND  WAVES  475 

Hence  the  index  of  refraction  (§667)  from  a  substance  of  dielec- 
tric constant  kl  to  another  of  dielectric  constant  &2  is  n  =\/fca/A;1. 
This  relation  has  also  been  verified  in  many  cases,  but  the  depend 
ence  of  n  on  the  wave-length  makes  the  test  difiicul  tin  other  cases. 
The  waves  started  from  Hertz's  oscillator  (§542)  are  plane 
polarized.  At  P'  there  is  an  alternating  electrostatic  force  in 
the  plane  of  the  diagram  and  an  alternating  magnetic  force  per- 


i  i  i 

f  T  f 


D 
Fio.  432. 


pendicular  to  that  plane.  These  together  constitute  the  vibra- 
tion in  the  front  of  the  wave  and  a  plane  polarized  wave  of  light 
is  similarly  constituted.  Thus  the  electromagnetic  theory  supple- 
ments the  wave  theory  stated  in  Light,  by  explaining  the  nature 
of  the  wave-motion. 

544.  Electric  Waves  along  Wires. — Fig.  432  shows  a  form  of 
Hertz  oscillator  as  modified  by  Lecher  to  show  electric  waves 
along  wires.  The  oscillations  produced  by  the  discharge  across 


Fia.  433. 

P,  act  by  static  induction,  and  produce  waves  which  traverse  the 
wires  C  and  D  and  are  reflected  back,  thus  forming  standing 
waves  by  interference  between  the  advancing  and  the  reflected 
waves  (§253) ,  similar  to  the  standing  waves  in  organ  pipes  (§609) . 
The  nodes  and  loops  can  be  detected  by  sliding  a  small  gap  along 
the  wires,  or  easier  by  a  device  due  to  Arons,  shown  in  Fig.  433. 
Arons  enclosed  the  two  wires  in  an  exhausted  glass  tube.  The 
loops  are  indicated  by  the  electrical  discharges,  while  the  nodes 
remain  dark. 

Seibt  has  arranged  a  beautiful  class-room  experiment  (Fig.  434) 
in  which  he  uses  a  Tesla  transformer  T  (§512)  as  oscillator,  and 


476 


ELECTRICITY  AND  MAGNETISM 


a  special  resonance  coil  CD  to  show  standing  waves.  The  verti- 
cal coil  CD  is  about  two  meters  high  and  consists  of  a  coil  of  silk- 
covered  wire  on  a  wooden  core.  Parallel  to  it  and  insulated  from 
it,  is  a  stretched  wire  MN.  The  nodes  and  loops  come  out 
brilliantly  in  a"darkened  room  as  indicated  in  Fig.  4346. 


Fio.  4340. 


FIG.  4346. 


545.  Detectors  of  Electric  Waves. — The  spark  gap,  which  Hertz 
used  so  successfully  in  his  investigations,  has  been  largely  re- 
placed by  more  sensitive  detectors.  (Cymoscope  has  been  pro- 
posed as  a  general  name  for  electric  wave  detectors.)  Almost 
every  effect  of  an  electric  current  has  been  used  in  these  de- 
tectors, such  as  heating,  magnetic,  electrolytic  and  resistance 
effects.  Only  two  of  these  detectors,  the  coherer,  and  the  crystal- 
rectifier  will  be  described  here.  The  reader  is  referred  to  special 
treatises  for  accounts  of  the  others. 

The  coherer,  in  the  form  given  to  it  by  Marconi,  consists  of  a 
small  glass  tube  TT'  Fig.  435,  in  which  there  are  two  silver  elec- 
trodes PP',  separated  by  a  small  quantity  of  loosely  packed 


ELECTRIC  OSCILLATIONS  AND  WAVES 


477 


FIG.  435. 


metal  filings.  A  mixture  of  95  per  cent,  nickel  and  5  per  cent. 
silver  filings  has  been  successfully  used  by  Marconi.  Marconi 
also  found  that  exhausting  the  tube  of  air  increased  the  relia- 
bility of  the  coherer.  The  action  of  the  coherer  depends  upon  a 
discovery  made  by  Branly  in  1900.  He  discovered  that  loosely 
packed  metal  filings,  which  offered  practically  infinite  resistance 
to  an  electric  current,  suddenly  acquire  good  conductivity  under 
the  action  of  an  electric  wave.  When  lightly  tapped  or  shaken,  the 
filings  again  lose  their  conductivity. 
The  generally  accepted  explanation 
is  that  the  small  filings  cohere  owing 
to  the  welding  action  of  the  infin- 
itesimal sparks  produced  by  the  electric  wave,  and  hence  the 
name  coherer  was  given.  The  coherer  is  not  selective  in  its 
action,  that  is,  it  responds  to  electric  waves  of  many  or  all 
lengths.  The  method  of  using  the  coherer  can  be  seen  from 
the  diagram  in  the  next  section. 

It  has  been  found  that  certain  crystals,  such 
as  silicon,  molybdenite,  and  carborundum, 
have  the  property  of  offering  much  greater 
resistance  to  the  passage  of  one-half  of  a  rap- 
idly alternating  electric  current  than  to  the 
other  half.  If  such  a  crystal  is  included  in 
ti16  circuit  of  a  receiving  antenna,  it  allows  the 
electric  oscillations  in  one  direction  to  pass,  but 
practically  cuts  out  the  oscillations  in  the  op- 
posite direction;  that  is,  it  "rectifies"  a  train 
of  rapidly  alternating  oscillations  into  a  train 
of  unidirectional  pulsations.  These  trains  of 
Fjo  436  electric  pulsations  can  be  detected  by  the 

clicks  in  a  telephone  circuit  which  is  connected 
with  the  antenna.     In  Fig.  436  is  shown  the  connections;  B 
represents  the  crystal  rectifier,  ABE  part  of  the  antenna  circui 
and  T  a  telephone. 

546.  Electric  Wave  Telegraphy  and  Telephony. — Since  1895, 
Marconi  has  developed  a  system  of  electric  wave  telegraphy, 
more  often  called  wireless  telegraphy,  for  transmitting  signals  to 
a  distance.  Using  very  powerful  oscillators  and  extremely  sensi- 
tive detectors,  Marconi  has  transmitted  messages  thousands  of 


478 


ELECTRICITY  AND  MAGNETISM 


miles.  This  system  has  been  particularly  successful  in  com- 
municating with  and  between  ships  at  sea.  Fig.  437  shows  a 
diagram  of  a  very  simple  electric  wave  telegraphic  arrangement. 
A  and  A!  are  high  vertical  lines,  called  antennae.  P  is  the  spark 
gap  of  the  sending  station,  C  is  the  coherer,  R  is  a  relay  operated 
by  any  current  through  C.  This  throws  in  the  battery  #2,  and 
excites  the  magnet  M  which  decoheres  C  by  tapping  it.  E  and 


A' 


Fro.  437. 

E  are  earth  connections.  Instead  of  the  coherer  C  and  sounder, 
more  reliable  and  more  sensitive  detectors  are  now  used. 

One  of  the  most  important  developments  in  electric  wave 
telegraphy  has  been  in  the  methods  of  producing  trains  of  un- 
damped waves.  The  difference  between  damped  and  undamped 
waves  is  indicated  by  Fig.  438a. 

The  waves  produced  by  an  ordinary  spark  discharge  are  as  we 
have  seen  damped  "waves  (see  Fig.  429).  The  simplest  means 
of  producing  undamped  waves  is  by  an  alternating  current 
dynamo;  but  the  design  of  such  a  machine,  so  as  to  have  a 
high  frequency  with  sufficient  output  of  electrical  energy,  is 
not  easy  because  there  is  a  limit  to  the  speed  at  which  an 
armature  can  be  safely  rotated.  An  entirely  different  method 
of  producing  undamped  electric  waves  is  that  first  patented  by 


ELECTRIC  OSCILLATIONS  AND  WAVES 


479 


Elihu  Thomson  in  1892.  This  is  shown  in  Fig.  4386.  The 
circuit  of  the  electric  generator  D  consists  of  a  coil  /  of  high 
inductance,  of  a  condenser  C,  an  inductance  L,  and  in  parallel 


FIG.  438o. 


o 


a  spark  gap  S.  When  a  spark  discharge  passes  across  S, 
persistent  electric  oscillations  are  set  up  in  the  circuit,  and 
hence  undamped  waves  are  sent  out.  Duddell  substituted  hard 
carbons  for  the  balls  S  of  the  spark  gap,  and  thus  produced 
the  "singing  arc."  Poulsen  has  developed 
the  method  still  further  by  using  a  hydro- 
carbon gas  about  the  arc,  making  the  posi- 
tive electrodes  of  copper,  etc.  Several  other 
systems  of  undamped  waves  are  also  in  use. 
A  great  advantage  of  undamped  waves  in 
wireless  transmission  is  that  more  power  can 
be  used  and  longer  distances  can  be  covered. 
The  most  striking  advantage  is,  however, 
that  it  makes  wireless  telephony  possible, 
if  the  frequency  of  the  undamped  waves  is  FIG.  4386. 
20,000  or  over.  This  frequency  produces  a  note  so  high  that  it 
is  not  audible.  The  lower  frequencies  of  the  voice  are  now 
superimposed,  by  introducing  a  telephone  transmitter,  as  varia- 
tions of  the  undamped  electric  waves,  and  these  variations  are 
reproduced  in  the  distant  receiving  apparatus,  without  inter- 
ference from  the  fundamental  waves.  In  1915,  a  notable 


480 


ELECTRICITY  AND  MAGNETISM 


advance  was  made  in  electric  wave  telephony  by  transmission 
of  speech  between  Washington,  D.C.  and  Eiffel  Tower  in  Paris. 

DIMENSIONS  OF  ELECTRICAL  UNITS 

547.  Kinds  of  Electrical  Units. — Three  kinds  of  eletrical  units 
have  been  defined  and  used  in  the  previous  sections,  the  electro- 
static units,  the  electromagnetic  units,  and  the  "practical"  units. 
The  practical  units  have  been  defined  as  multiples  of  the  elec- 
tromagnetic units,  the  multiples  being  chosen  so  as  to  make  units 
of  convenient  sizes  for  calculations  in  the  technical  application 
of  electricity.  The  electrostatic  and  electromagnetic  units  are 
both  "absolute  units"  that  is,  are  based  by  definitions  on  simple 
relations  to  the  fundamental  units,  the  units  of  length,  mass  and 
time  (§150).  The  particular  absolute  system  long  universally 
used  in  electricity  and  magnetism  is  that  based  on  the  centimeter, 
the  gram  and  second,  or  the  c.g.s.  system  (§150).  The  follow- 
ing table  shows  the  relations  of  the  practical  and  absolute 
electrical  units. 

ELECTRICAL  UNITS 


Unit  of 

Name  of 
practical 
Unit 

Value  of  practical'  Unit 

in  C.G.S. 
E.M.U. 

in  C.G.S. 
E.S.U. 

Current  .  . 

Ampere 
Coulomb 
Volt 
Ohm 
Farad 
Henry 

io-1 

10-1 
10" 
10° 

io-» 

10" 

3.10° 
3.10' 
1/(3X109) 
1/C9X1011) 
9.1011 

T/Toxio11) 

Quantity  ., 

Electromotive  force  
Resistance  

Caoacitv 

Inductance  

The  establishment  and  universal  use  of  an  absolute  system  of 
units  in  electricity  and  magnetism  has  contributed  much  to  the 
progress  of  the  science  both  in  its  theory  and  in  its  applications. 
The  relations  of  the  units  of  electric  quantity,  current,  potential, 
etc.,  to  the  units  of  energy  and  power  are  clear  and  direct  in  an 
absolute  system.  Thus  the  product  of  the  number  of  units  of 
current  and  of  potential  gives  directly  the  number  of  units  of 
power  or  activity,  no  arbitrary  constants  entering  into  the  calcu- 
lations. The  advantage  of  this  simplicity  is  evident.  Again  the 


ELECTRICAL  UNITS 


481 


study  of  the  dimensions  of  the  units  (§151),  has  led  to  a  clearer 
view  of  the  nature  of  electrical  and  magnetic  quantities,  and  of  the 
relations  of  electrical  phenomena  to  other  phenomena.  Thus  the 
comparison  of  the  dimensions  of  the  electrostatic  and  electromag- 
netic units  suggested  to  Maxwell  important  similarities  of  the 
electrical  and  optical  effects,  and  contributed  much  to  Maxwell's 
electromagnetic  theory  of  light  (§543).  This  last  theory  was 
again  a  starting-point  for  speculations  which  resulted  in  Hertz's 
epoch-making  experiments  on  electric  waves  and  their  prop- 
erties (§542).  The  dimensions  of  electrical  and  magnetic  units 
thus  have  a  greater  importance  than  that  of  translating  results 
from  one  absolute  system  to  another  (§151). 

548.  Dimensions  of  Electrical  Units. — The  following  table  gives  the 
dimensions  of  five  of  the  more  usual  electrostatic  and  electromagnetic 
units. 


Name 


Symbol 


Electrostatic 


Electromagnetic 


Electric  quantity 
Magnetic  quantity 
Magnetic  field 
Current 
Potential  or 
Electromotive  force 


m 
H 
I 

E 


The  method  of  deriving  the  above  dimensions  from  the  definitions  is 
shown  by  the  following  examples. 

Electrostatic  Unit  of  Quantity.  We  have  by  definition  (§401)  q^r^/Fk 
Using  the  dimensions  of  r  and  F,  we  get  [q\  =  [L*T-lMlk%\.  In  this  k  is 
the  specific  inductive  capacity  or  dielectric  constant  (§401),  a  quantity 
arbitrarily  assumed  as  unity  for  air  but  of  undetermined  dimensions. 

Electrostatic  Unit  of  Current.  By  definition  (§429)  [/]— g/*.  Sub- 
stituting the  dimensions,  we  get  [I]  =  [L$T~*Mlk$]. 

The  starting-point  in  the  electromagnetic  system  is  the  definition  of  unit 
magnetic  pole  (§372),  m=rv^i,  where  /*  is  the  magnetic  permeability 
(§491),  a  quantity  arbitrarily  assumed  as  unity  for  air,  but  of  undetermined 
dimensions.  From  this  we  get  the  dimensions  of  [m]=»[L^7T~1M^i]. 
From  the  relation  that  F,  the  force  at  a  point  in  a  magnetic  field  is  mil, 
we  get  H  =  F/m.  The  dimensional  equation  for  intensity  of  magnetic 
field  is  thus  [B]-[L-*T'- 
31 


482  ELECTRICITY  AND  MAGNETISM 

Electromagnetic  Unit  of  Current.  The  strength  of  magnetic  field  at  the 
center  of  a  circular  coil  of  Radius  r,  and  carrying  a  current  7,  is  H=*2nl/r 
(§428);  substituting  dimensions,  we  get  [/]-[#]  [^-[Lir-'Afi/i-l]. 

Electromagnetic  Unit  of  Quantity.     From  the  relation  q=°>Itt  we  get 


Comparing  the  electrostatic  and  electromagnetic  units  of  quantity,  we 
get  the  ratio  [LiT-lMld]+[L*Mlp-*}-[LT-litp*].  But  LT-*  is  a 
velocity  (§150).  This  velocity  "v"  also  appears  in  the  ratios  of  the 
other  units,  though  not  always  as  the  first  power.  If  an  electric  quantity 
is  measured  in  air  both  electrostatically  and  eleotromagnetically,  then  both 
k  and  p  are  assumed  as  unity  and  the  value  of  this  velocity  "v"  can  be 
determined.  This  was  first  done  by  Weber  and  Kohlrausch  in  1856,  by 
determining  the  electric  quantity  in  a  condenser  from  its  electrostatic 
capacity  and  potential  (§410),  and  also  by  discharging  the  same  quantity 
through  a  ballistic  galvanometer  (§439).  They  obtained  the  value 
v  =-310,704,000  meters  per  second.  This  number  is  within  limits  of 
error  the  same  as  the  velocity  of  light.  This  equality  has  been  established 
by  numbers  of  later  determinations.  The  close  connection  between  the 
velocity  of  light  and  the  ratio  of  the  electrostatic  and  electromagnetic 
units  confirmed  Maxwell  in  the  theory  that  light  is  a  phenomenon  of  the 
same  nature  as  that  of  electromagnetic  actions  (§543). 

If  we  assume  the  equality  of  the  two  units  of  quantity,  without  assuming 
k  and  ft  as  unity,  we  get  directly  that  "v"  —  l/Vkp.  This  is  a  very  signifi- 
cant relation,  and  has  been  the  subject  of  much  experiment  but  has  been 
only  partially  confirmed. 

References 

GILBERT,  SIR  WILLIAM.     "De  Magnete,"  translation  by  P.  F.  Mottelay. 

FRANKLIN,  BENJAMIN.    Letters  on  Electricity. 

FARADAY,  MICHAEL.     Experimental  Researches  in  Electricity. 

MAXWELL,  JAMES  CLERK.     Treatise  on  Electricity  and  Magnetism. 

HERTZ,  HEINRICH.     Electric  Waves,  translation  by  Jones. 

THOMSON,  J.  J.     Conduction  of  Electricity  through  Gases. 

The  above  are  the  electrical  "classics."  The  present  interest  in  the 
writings  of  Gilbert  and  Franklin  is  principally  historical.  Faraday's 
papers  are  invaluable  in  giving  an  insight  into  the  methods  of  thought 
and  work  of  the  greatest  experimenter  in  electricity.  An  inex- 
pensive edition  of  part  of  Faraday's  "Researches  in  Electricity"  is  now 
available.  Maxwell's  Electricity  and  Magnetism  is  a  work  to  be  classed 
with  Newton's  Principia  as  bringing  an  epoch  in  science;  it  uses  very 
advanced  mathematical  methods.  Hertz's  and  J.  J.  Thomson's  papers 
are  the  classics  to  the  two  great  lines  of  recent  electrical  research. 

FOSTER  AND  PORTER.     Electricity  and  Magnetism. 

STARLING,  S.  G.     Electricity  and  Magnetism. 

POYNTINQ  AND  THOMPSON.     Electricity  and  Magnetism, 


REFERENCES  483 

BROOKS  AND  POYSER.     Magnetism  and  Electricity. 

HADLEY.     Magnetism  and  Electricity  for  Students. 

RICHARDSON,  S.  S.     Magnetism  and  Electricity. 

THOMPSON,  S.  P.     Electricity  and  Magnetism. 

Excellent  text-books  covering  the  same  subjects,  the  last  four  being 
more  elementary  than  the  first  three. 

JEANS.     Magnetism  and  Electricity. 

THOMSON,  J.  J.     Mathematical  Theory  of  Electricity  and  Magnetism. 

Two  good  books  giving  an  introduction  to  the  mathematical  theory  of 
electricity. 

EWING.     Magnetic  Induction  in  Iron  and  Other  Metals. 
The  best  book  on  experimental  magnetism. 

CAMPBELL,  N.  R.     Modern  Electrical  Theory. 

CAMPBELL,  N.  R.     Principles  of  Electricity.     (People's  Library). 

FOURNIER  d'ABBE  E.  E.     The  Electron  Theory. 

DE  TUNZELMANN,  G.  W.  Electrical  Theory  and  the  Problem  of  the  Universe. 
These  books  give  interesting  discussions  of  the  explanation  of 
electrical  phenomena  on  the  electron  theory.  They  are  largely  or 
wholly  non-mathematical. 

LODGE,  OLIVER  J.     Modern  Views  of  Electricity. 

A  very  suggestive  book  by  a  leader  in  modern  physics. 

FLEMING,  J.  A.     Principles  of  Electric  Wave  Telegraphy. 

FLEMING,  J.  A.     Radio-telegraphy  and  Radio-telephony. 

FLEMING,  J.  A.     Waves  in  Water,  Air  and  Ether. 

The  first  book  is  our  most  complete  treatise  on  wireless  telegraphy, 
and  gives  the  advanced  mathematical  discussions  along  with  experi- 
mental facts.  The  second  book  presents  the  same  subject  in  much 
shorter  form,  and  is  not  mathematical.  The  third  book  is  based  on  a 
series  of  popular  scientific  lectures  on  electric  waves. 

PIERCE,  G.  W.     Wireless  Telegraphy. 

STANLEY,  R.     Wireless  Telegraphy. 

Excellent  books  presenting  the  subject  as  completely  as  possible 
without  advanced  mathematics. 

CARHART,  H.  S.     Primary  Batteries. 

MORSE,  H.  W.     Storage  Batteries. 

LE  BLANC.     Electrochemistry. 

NERNST,  W.     Theoretical  Chemistry. 

LODGE,  OLIVER  J.     Lightning  Conductors  and  Lightning  Guards. 
The  most  satisfactory  book  on  an  obscure  subject. 

THOMPSON,  S.  P.     Dynamo-electric  Machinery. 

Bulky  and  containing  many  details  of  only  special  interest,  but  also 
giving  an  excellent  presentation  of  the  physics  of  dynamos  and  motors. 

FLEMING,  J.  A.     The  Alternate  Current  Transformer. 

A  clear  presentation  of  electromagnetic  induction  in  the  transformer. 

ENCYCLOPEDIA  BRITANNICA,  eleventh  edition. 

The  electrical  articles  are  by  masters  and  form  in  themselves  a  very 
valuable  and  complete  reference  library  in  electricity  and  magnetism. 


484  ELECTRICITY  AND  MAGNETISM 

Problems 

1.  Find  the  intensity  of  field  at  a  point  40  cm.  from  a  magnet  in  the  per- 
pendicular bisector  of  the  line  joining  the  poles  of  the  magnet  6  cm. 
long  and  of  pole  strength  160  e.m.u.     Calculate  the  force  on  a  pole 
of  4-80  e.  m.  u.  if  placed  at  the  point.       Ans.  .0148  e.m.  u.;  1.19  dynes. 
2.  A  magnet  NS  30  cm.  long  is  held  vertically;  each  pole  has 
Magnetic  a  strength  of  9  units;  what  is  the  force  on  a  unit  pole  at 

Field.  a  point  20  cm.  horizontal  distance  from  the  upper  pole? 

What  is  the  horizontal  component  of  this  force? 

3.  The  point  P  is  on  the  perpendicular  bisector  of  a  magnet  NS,  at  a 
distance  of  30  cm.  from  NS.     A  pole  of  strength  8  at  P  is  acted  on 
with  a  force  of  3  dynes.     Find  the  moment  of  the  magnet  NS. 

4.  To  hold  a  magnetic  needle  NS  at  an  angle  of  60°  with  the  earth's  field 
requires  a  torque  of  0.6  dynes  acting  at  a  lever  arm  of  2  cm.;  the  hori- 
zontal intensity  of  the  earth's  field  is  0.2;  what  is  the  moment  of  the 
magnet? 

6.  A  short  bar  magnet  is  placed  with  its  axis  perpendicular  to  the  magnetic 
meridian,  and  with  the  line  of  the  axis  passing  through  the  center  of  a 
compass  needle.  At  a  station  X,  the  compass  needle  is  deflected 
through  an  angle  <f>,  when  the  center  of  the  magnet  is  40  cm.  from  the 
center  of  the  needle.  At  a  station  Y,  the  distance  from  magnet  to 
needle  is  35  cm.  for  the  same  deflection  <j>.  Compare  the  horizontal 
intensities. 

6.  The  horizontal  intensity  of  the  earth's  field  at  Indianapolis  is  0.2,  and 
at  Minneapolis  it  is  0.18;  if  a  magnetic  needle  makes  100  vibrations  at 
Indianapolis,  what  will  its  period  be  at  Minneapolis? 

7.  To  deflect  a  suspended  magnet  through  an  angle  of  20°  from  the  mag- 
netic meridian  requires  180°  of  torsion  in  the  wire  suspension;  how 
many  degrees  of  torsion  must  be  given  the  suspension  to  produce  a 
deflection  of  45°  from  the  magnetic  meridian? 

8.  A  horizontal  magnetic  needle  makes  40  oscillations  per  minute  at  a 
place  where  the  dip  is  70°,  and  50  oscillations  per  minute  where  the  dip 
is  60°.     The  total  intensity  at  the  first  place  is  0.6;  what  is  it  at  the 
second  place? 

9.  The  center  of  a  short  bar  magnet  is  at  the  corner  A  of  a  square  A  BCD 
and  its  axis  is  in  line  with  the  side  AB.     The  moment  of  the  magnet  is 
400,  and  the  length  of  one  side  of  the  square  is  60  cm.;  find  the  intensity 
of  the  magnetic  field  at  the  corners  B  and  D,  due  to  the  bar  magnet. 

10.  Two  small  spheres,  each  weighing  1  decigram,  having  equal  charges, 

are  suspended  from  the  same  point  by  silk  fibers  80  cm.  long.     If  the 

spheres  are  kept  8  cm.  apart  by  repulsion,  what  is  the  charge  on  each? 

11.  Two  charges  +90  and  —40  are  30  cm.  apart.     Find 

Electrostatic  the  intensity  of  field  at  a  point  in  the  line  joining  them 

Fields.  60  cm.  from  the  negative  and  90  cm.  from  the  positive 

charge,  and  calculate  the  force  on  a  charge  of  +20  if 

placed  at  this  point. 


PROBLEMS  485 

12.  Two  small  charged  spheres  repel  each  other  with  a  force  of  10  dynes 
when  2  cm.  apart.     If  the  charge  on  one  of  the  spheres  is  doubled,  and 
the  distance  between  the  spheres  is  doubled,  what  is  the  repulsion? 

13.  What  work  is  done  in  carrying  a  charge  of  10  units  from  a  point  where 
the  potential  is  25  to  a  point  where  it  is  40? 

Ca      ..        14.  A  Leyden  jar  of  capacity  10  e.s.u.  is  raised  from  a  poten- 
tial -10  e.s.u.  to  a  potential  +15  e.s.u.     Calculate  the 
work  required. 

15.  Given  two  spheres  of  radii,  3  cm.  and  8  cm.,  how  will  a  charge  of  66 
units  distribute  itself  over  them  if  they  are  connected  by  a  fine  wire? 

16.  What  is  the  charge  on  a  spherical  drop  of  water  2  mm.  in  diameter, 
where  the  electric  potential  is  100?     Two  such  charged  drops  unite  to 
form  a  single  spherical  drop;  assuming  no  charge  is  lost,  what  is  the 
potential  of  the  resulting  drop?     If  three  drops  thus  unite,  what  is  the 
final  potential? 

17.  A  Leyden  jar  1/4  cm.  thick  is  3  cm.  in  radius  and  9  cm.  high.     Find  its 
capacity  if  the  dielectric  constant  for  glass  is  6.     Find  charge  on  each 
plate  when  p.  d.  is  15  e.s.u. 

18.  A  condenser  of  10  plates,  each  20  cm.  X30  cm.  has  0.4  mm.  of  air 
between  each  pair  of  plates.     Find  the  capacity. 

19.  Two  plate  condensers  are  joined  in  parallel.     One  is  a  15  plate  air  con- 
denser, each  plate  11  cm.  long  and  5  cm.  broad,  3  mm.  apart;  the  other 
a  mica  condenser  of  10  plates,  22  cm.  long,  15  cm.  broad,  0.5  mm.  apart, 
specific  inductive  capacity  of  mica  being  8.     Find  the  capacity. 

20.  Two  concentric  spheres  of  radii  10  cm.  and  10.3  cm.  are  separated  by 
air  and  are  charged  to  difference  of  potential  of  50  volts.     Find  charge. 

21.  A  pair  of  circular  plates  of  radii  10  cm.  each  are  2  mm.  apart  in  air. 
They  are  charged  to  a  difference  of  potential  of  20  and  are  then  connected 
to  the  plates  of  an  uncharged  condenser  and  the  difference  of  potential 
falls  to  3.     Find  the  capacity  of  this  condenser. 

22.  Find  the  capacity  of  a  plate  condenser  made  of  two  rectangular  con- 
ductors 32  cm.  long  and  22  cm.  broad,  0.2  cm.  apart  in  air. 

23.  If  the  air  be  replaced  by  0.2  cm.  sheet  of  glass  of  dielectric  constant  7, 
find  the  charge  on  each  plate  when  the  difference  of  potential  is  20 
e.s.u. 

24.  Find  the  work  in  ergs  required  to  charge  an  insulated  metal  ball  of 
radius  5  cm.  with  20  e.s.u.  of  electricity. 

25.  A  circular  coil  of  30  cm.  diameter  has  20  turns. 
Magnetic  Fields  Compute  the  intensity  of  the  magnetic  field  at  the 

of  Currents.  center  when  a  current  of  10  amperes  flows  through 

the  coil. 

26.  Find  the  field  strength  16  cm.  from  the  center  of  a  coil  in  the  line  of  its 
axis  if  the  coil  carry  0.5  amp.  and  be  24  cm.  in  diameter. 

27.  Find  force  on  a  pole  of  30  e.m.u.  if  placed  at  center  of  coil  in  problem  26. 

28.  Calculate  current  which  will  deflect  a  tangent  galvanometer  45°,  if  the 
galvanometer  consists  of  a  coil  18  cm.  in  diameter,  of  7  turns  of  wire, 
set  up  in  a  field  of  0.198  lines  per  cm2. 


486  ELECTRICITY  AND  MAGNETISM 

29.  The  coil  of  a  tangent  galvanometer  is  34  cm.  in  diameter  and  carries  a 
current  of  15  amperes;  what  is  the  torque  on  a  needle  of  moment  1.5  at 
the  center? 

30.  A  coil  of  a  tangent  galvanometer  is  to  have  10  turns;  what  should  the 
radius  of  the  coil  be,  so  that  the  tangent  of  the  angle  of  deflection  of  the 
needle  gives  directly  the  current  in  amperes,  at  a  station  where  the 
intensity  of  the  earth's  field  is  0.19? 

31.  A  circular  coil  was  placed  at  right  angles  to  the  magnetic  meridian. 
The  number  of  oscillations  of  a  small  magnetic  needle  at  the  center  was 
counted  (a)  when  there  was  no  current  in  the  coil;  (6)  when  a  current 
t"i  was  sent  through  the  coil;  (c)  when  a  current  ia  was  used.     For  (a) 
there  were  40  oscillations  per  minute;  for  (6)  30  oscillations  per  minute; 
and  for  (c)  20  oscillations  per  minute;  what  was  the  relative  strengths 
of  the  currents  t\  and  ia? 

32.  A  slender  solenoid  has  a  length  of  50  cm.  and  has  300  turns  of  wire; 
what  is  the  field  at  the  center  when  the  current  in  the  coil  is  7  amperes? 

33.  Calculate  the  current  which  produces  a  magnetic  field  in  the  middle  of 
a  slender  solenoid  equal  to  the  earth's  field  of  0.6,  the  solenoid  being 
80  cm.  long,  and  having  400  turns. 

W    k      a     34.  A  current  of  6  amperes  flows  for  4  min.  in  a  circuit  of 
Ohm's  Law  12  ohms  resistance;  what  is  the  e.m.f.  required?     What 

is  the  total  work  done  in  ergs,  also  in  joules?     What  is 
the  activity  or  power  in  watts? 

36.  Three  electromagnets  of  resistances  50,  76  and  11  ohms  respectively  are 
joined  in  multiple  arc,  and  a  total  current  of  2  amperes  flows  through 
the  three;  what  is  the  current  in  each? 

36.  The  total  resistance  of  a  circuit  is  80  ohms,  and  on  introducing  an 
addition  wire  the  resistance  is  66  ohms;  what  is  the  resistance  of  the 
added  wire? 

37.  A  circuit  has  three  branches  of  50,  30  and  10  ohms.     A  fourth  branch 
is  put  in  so  that  the  total  resistance  is  2  ohms.     What  is  the  resistance 
of  the  fourth  branch? 

38.  A  generator  delivers  100  amperes  at  110  volts;  what  is  the  power  in 
kilowatts  and  what  in  H.  P.? 

39.  The  resistance  of  a  galvanometer  is  126  ohms,  and  a  shunt  of  14  ohms 
is  put  in;  what  is  the  resistance  of  the  shunted  galvanometer? 

40.  By  experimenting  with  a  Weston  ammeter  it  was  found  that  0.00013 
amperes  through  the  coil  gave  one  unit  scale-deflection.     If  the  resist- 
ance of  the  coil  circuit  be  5.60  ohms,  what  must  be  that  of  the  shunt  so 
that  1  ampere  in  the  external  circuit  will  give  1  unit  scale  deflection? 

41.  It  is  desired  to  supply  600  incandescent  lamps,  in  parallel,  with  1/2 
amp.  each,  at  110  volts  potential  difference  between  the  lamp  terminals. 
If  the  drop  in  the  line  be  2.2  volts  what  is  the  resistance  of  the  line  and 
how  much  power  is  lost  in  it?     How  much  power  must  be  generated 
and  what  voltage? 

42.  A  car  is  lighted  by  five  lamps  of  220  ohms  resistance  each,  joined  in 
series.     What  is  the  total  resistance  of  the  lamps?    If  the  difference 


PROBLEMS  487 

of  potential  between  the  ends  of  the  lamp  circuit  be  550  volts,  what 
current  flows  through  the  lamps?  What  power  is  expended  in  this 
circuit  and  at  9  c.  per  kilowatt  hr.,  what  does  it  cost  to  light  a  car 
for  one  hour? 

43.  If  the  motive  circuit  of  a  snow  sweeper  take  50  amp.  (at  550  volts)  and 
the  broom  motors  take  80  amp.,  find  the  total  power  consumed  in  the 
car  if  the  two  circuits  be  in  parallel  across  550  volt  mains.    Find  cost 
per  hour  at  9  c.  per  kilowatt  hr. 

44.  A  single  lighting  circuit  carries  3  groups  of  lamps  in  multiple,  the  groups 
being  100  feet  apart,  and  the  nearest  group  being  500  feet  from  the 
generator.     Each  group  takes  5  amperes,  and  the  resistance  of  the  line 
0.1  ohm  per  1000  feet.     The  potential  of  the  generator  is  112  volts. 
Find  the  potential  of  each  group  of  lamps. 

45.  The  distance  from  power  house  to  library  is  650  meters,  there  are  200 
55-watt  lamps  in  library.     The  e.m.f .  at  dynamo  is  465  volts.     Allowing 
5  per  cent,  drop  of  potential  in  the  line,  what  must  be  the  resistance  of 
the  line.     Taking  the  specific  resistance  of  copper  as  1.5X10-*  ohms 
per  cm./cm.a,  what  is  the  cross-section  of  the  wire?     Calculate  the 
watts  on  full  load  for  this  circuit.     What  must  the  H.  P.  of  the  engine  be 
to  carry  this? 

46.  Find  the  resistance  in  legal  ohms  of  a  tube  of  mercury  at  0°C.,  1  meter 
long,  and  1  cm.  in  diameter. 

47.  The  resistance  of  a  certain  firm's  copper  wire  1  foot  long  and  a  mill 
(one  thousandth  of  an  inch)  in  diameter  is  10.7  ohms.     What  is  its 
specific  resistance  in  ohms  per  cm. /cm.2?     1  inch  —  2.540  cm. 

48.  The  specific  resistance  of  copper  is  1.5  X  10~6  and  aluminum  is  3.2  X  10~6; 
with  copper  at  18  c.  per  pound,  what  must  be  the  price  of  aluminum  to 
compete  as  an  electrical  conductor? 

49.  Given  3  cells  of  1.4  volts  and  0.8  ohms  resistance  each,  find  resistance  of 
the  battery  if  the  cells  be  connected  in  series  and  calculate  the  current 
through  an  external  resistance  of  9  ohms. 

50.  Find  the  resistance  of  the  above  battery  if  cells  be  in  parallel  and  also 
the  current  when  the  external  resistance  is  9  ohms. 

61.  Given  20  cells,  each  with  an  e.m.f.  of  1.7  volts,  and  an  internal  resistance 
of  3  ohms.     Calculate  the  current  in  the  following  cases: 

(a)  External  resistance  72  =  100  ohms,  cells  in  series. 

(b)  R  =100  ohms,  cells  in  parallel. 

(c)  R  =20  ohms,  cells  in  series;  also  in  parallel. 

(d)  R  =20  ohms,  4  parallel  rows  of  5  cells  in  series. 

(e)  Arrangement  for  maximum  current  through  20  ohms. 

52.  A  "milli-ammeter,"  which  is  to  be  used  as  a  voltmeter,  indicates  .005 
amperes  for  a  scale  division,  and  has  a  resistance  of  40  ohms.     There  are 
50  scale  divisions.     What  resistance  in  series  with  the  instrument  will 
enable  it  to  be  used  for  measurements  up  to  300  volts- 

53.  It  is  required  to  generate  1000  calories  of  heat  per  minute  in  a  circuit, 
the  e.m.f.  at  the  terminals  of  the  circuit  being  110  volts;  what  resistance 
must  the  circuit  have? 


488  ELECTRICITY  AND  MAGNETISM 

54.    The  same  current  flows  through  a  platinum  wire  25  cm.  long 

Joule's  and  0.5  mm.  diameter,  and  through  a  copper  wire  500 

Law.  cm.  long  and  0.6  mm.  diameter.     What  are  the  relative 

heat   quantities    developed    in    these   wires?     (Sp.   R.   of 

Cu  =1.5X10-'.   Sp.  R.  of  Pt  =8.9X10-8). 

66.  A  uniform  current  flows  for  10  minutes  and  deposits  4  grams  of  silver; 

calculate  the  current. 

56.  How  much  copper  can  a  dynamo  giving  30  amperes  deposit  in  an  hour? 
Ele  tr  1    '        ^'  ^ow  manv  cubic  centimeters  of  hydrogen  at  0°C.  and 
76  cm.  Hg.  pressure  can  a  current  of  30  amperes  pro- 
duce by  the  decomposition  of  acidulated  water  in  an  hour?     (Sp.  gr.  of 
H  =00008.9  at  0°C.  and  76  cm.  Hg.  pressure.) 

-..          .        58.  An  iron  anchor  ring  has  20  cm.  mean  diameter,  and  a  cross- 

,    .     ^  section  of  18  sq.  cm.    The  coil  has  600  windings  and  carries 

10  amperes.     How  great  is  the  magnetic  induction  5? 

How  many  magnetic  lines  are  produced?     (The  permeability  fj.  =200.) 

_.  59.  Show  that  Lenz's  law  and  Fleming's  rule  lead  to 

Electromagnetic  ,.      ,.      .          .    ,       .   ' 

T    .     ..  the  same  direction  for  an  induced  current  in  a  con- 

Induction.  ,  .  A.    -  .  . 

ductor  moved  across  a  magnetic  field. 

60.  The  diameter  of  a  circular  coil  is  30  cm.  and  the  resistance  is  0.1  ohm. 
Find  the  quantity  of  electricity  in  coulombs  which  will  flow  in  the  ring 
when  revolved  from  a  position  at  right  angles  to  a  magnetic  field  to  a 
position  parallel  to  the  field.     H  =20. 

61.  A  circular  coil  40  cm.  in  diameter  and  with  100  turns  is  rotated  five 
times  per  second  about  a  vertical  diameter  as  axis.     Find  the  maximum 
e.m.f.  induced.     The  horizontal  component  of  the  field  is  0.2. 

62.  If  the  angle  of  dip  is  70°,  what  is  the  maximum  e.m.f.  induced  when  the 
above  coil  is  rotated  about  a  horizontal  axis  parallel  to  the  horizontal 
component  of  the  field  ten  times  per  second? 

63.  Calculate  the  e.m.f.  induced  in  a  car  axle  length  120  cm.  and  with  a 
horizontal  linear  velocity  of  25  meters  per  second,  where  the  total  in- 
tensity of  the  field  is  0.6  and  the  angle  of  dip  is  70°. 

64.  Calculate  the  number  of  revolutions  per  second  which  must  be  given 
to  a  disk  of  60  cm.  diameter  to  produce  an  e.m.f.  of  5  volts  between  the 
center  and  the  periphery  of  the  disk,  the  axis  of  the  disk  being  parallel 
to  the  field,  and  the  field  being  uniform  and  of  strength  10,000. 

65.  A  copper  disk  10  cm.  in  radius  rotates  about  a  vertical  axis  with  2000 
r.  p.  m.     Given  the  horizontal  component  of  the  earth's  magnetic  field 
as  0.2,  and  the  dip  as  70°,  find  the  e.m.f.  in  volts  between  the  center  and 
edge  of  the  disk. 

66.  Draw  a  figure  showing  the  directions  of  the  induced  currents  in  the  disk 
of  a  pendulum  swinging  between  the  poles  of  a  magnet  across  the  field. 
How   should  the  disk  be  laminated  to  make  the  induced  currents  a 
minimum? 

67.  A  rectangular  coil  10  cm.  X  12  cm.  can  rotate  about  a  vertical  axis 
which  bisects  the  12  cm.  sides.     A  current  of  3  amperes  flows  through  the 
coil,  and  the  horizontal  intensity  of  the  magnetic  field  is  0  2.     What  is  the 
torque  (moment  of  force),  when  the  coil  is  at  right  angles  to  the  field? 


CONDUCTION  OF  ELECTRICITY  THROUGH 
GASES  AND  RADIO-ACTIVITY 

BY  R.  K.  McCLUNG,  D.Sc.,  F.  R.  S.  C. 

Assistant  Professor  of  Physics  in  the  University  of  Manitoba 
CONDUCTION  OF  ELECTRICITY  THROUGH  GASES 

549.  Introduction. — Air,  as  well  as  other  gases,  under  normal 
conditions   is   almost   a   perfect   non-conductor   of   electricity. 
When  a  difference  of  potential  is  established  between  two  points 
in  a  gas  the  gas  is  in  a  state  of  strain,  as  has  been  explained  in  a 
former  paragraph  (§398).     This  strain  increases  with  increase 
of  potential  until,  when  a  certain  potential  is  reached,  the  air  is 
no  longer  able  to  withstand  the  strain  and  breaks  down  and  a 
discharge  passes.     A  momentary  current  of  electricity  is  thus 
produced  through  the  gas.     To  produce  such  a  discharge  a 
comparatively  large  potential  is  required,  several  thousand  volts 
being  necessary  to  produce  a  spark  of  1  cm.  length  in  air  at 
atmospheric  pressure.     The  potential  necessary  to  produce  a 
discharge  depends  upon  the  shape  of  the  electrodes  and  the 
nature  and  pressure  of  the  gas. 

550.  Effect  of  Pressure  of  a  Gas  on  the  Discharge. — If  two 
metal  electrodes  are  inserted  in  the  ends  of  an  air-tight  glass  tube, 
such  as  shown  in  Fig.  439,  filled  with  air  at  atmospheric  pressure, 
and  if  sufficient  voltage  is  ap- 

plied  to  the  electrodes  the  dis-    -mn-o— £| 
charge  ordinarily  obtained  in 
air  will  be  observed.     If  the  Flo  439 

air    be    gradually  exhausted 

from  the  tube  the  discharge  will  pass  with  greater  and  greater 
ease  as  the  pressure  is  diminished,  until  a  certain  minimum 
pressure  is  reached,  and  if  the  exhaustion  be  carried  beyond 
this  point  the  voltage  necessary  to  produce  a  discharge  will 
increase  somewhat  rapidly,  until  at  the  lowest  pressure  ob- 
tainable it  will  be  impossible  to  cause  a  discharge  to  pass  at  all. 

489 


490       CONDUCTION  OF  ELECTRICITY  THROUGH  GASES 

The  pressure  corresponding  to  this  minimum  potential  is  called 
the  critical  pressure  and  varies  with  the  distance  between  the 
electrodes. 

As  the  pressure  is  gradually  diminished  below  atmospheric 
pressure  the  appearance  of  the  discharge  changes  very  much. 
At  first  the  spark  becomes  more  regular  and  uniform  between  the 
electrodes,  then  broadens  out  and  assumes  a  fuzzy  appearance  of 
a  bluish  color.  When  a  pressure  of  about  half  a  millimeter  is 
reached  the  discharge  assumes  a  very  marked  appearance,  which 
is  shown  in  Fig.  440.  The  surface  of  the  negative  electrode  or 


Fio.  440. 


cathode  is  covered  by  a  very  thin  layer  of  luminosity;  next  to  this 
is  a  dark  space  which  is  called  the  Crookes  dark  space;  immediately 
beyond  this  dark  space  is  a  luminous  part  called  the  negative  glow, 
and  then  beyond  this  again  is  a  second  dark  region,  sometimes 
called  the  Faraday  dark  space.  Between  this  and  the  anode 
there  is  a  luminous  region  which  goes  under  the  name  of  the 
positive  column.  Under  certain  conditions  of  current  and  pres- 
sure the  positive  column  shows  alternately  dark  and  light  spaces 
which  are  called  stria.  The  proportion  of  the  space  between  the 
electrodes  occupied  by  each  of  these  sections  of  the  discharge 
depends  upon  the  distance  between  the  electrodes.  Any  increase 
in  this  distance  beyond  a  few  centimeters  causes  an  increase  in 
the  length  of  the  positive  column  but  no  increase  in  the  negative 
glow  or  dark  space.  Similar  phenomena  occur  in  other  gases 
besides  air. 

561.  Cathode  Rays. — When  the  pressure  in  such  a  discharge 
tube  is  lowered  to  the  neighborhood  of  a  hundredth  of  a  milli- 
meter, a  new  phenomenon  makes  its  appearance.  The  positive 
column  begins  to  disappear  and  a  bright  phosphorescence  appears 
on  the  sides  of  the  tube.  This  phosphorescence  appears  to  be 
produced  by  radiations  or  streams  of  very  minute  particles  issuing 
normally  in  straight  lines  from  the  cathode.  They  are,  con- 
sequently, called  cathode  rays,  and  possess  remarkable  properties. 

If  a  magnet  is  brought  close  to  the  tube  the  rays  are  deflected 


CATHODE  RAYS 


491 


from  their  original  path.  A  solid  body  placed  inside  the  tube  in 
the  path  of  the  rays  casts  a  well-defined  shadow.  If  the  rays  be 
concentrated  upon  a  solid  body  inside  the  tube,  such  as  a  plati- 
num plate,  it  may  be  heated  even  to  incandescence. 

One  of  the  most  important  properties  of  the  cathode  rays  is 
that  they  carry  a  negative  charge  of  electricity.  This  was  originally 
proved  by  Perrin  and  his 
method  was  later  modified 
by  J.  J.  Thomson.  A  diagram 
of  the  apparatus  used  in  the 
latter  experiment  is  shown  in 
Fig.  441.  A  was  the  cathode 
and  B  the  anode.  The  cathode 

_    . 

rays  from  A  passed  into  the 
larger  part  of  the  tube  through  a  hole  in  B  and  fell  upon 
the  glass  at  a  point  C.  A  side  tube  contained  two  coaxial 
metal  tubes.  The  outer  one  E  had  a  slit  in  the  end  and  was 
connected  to  earth.  This  shielded  the  inner  tube  from  any 
stray  electric  effects.  The  inner  tube  D  had  a  slit  opposite 
that  in  E  and  was  insulated  from  E  and  connected  to  an 
electrometer.  When  the  cathode  rays  were  allowed  to  fall  upon 
the  glass  bulb  the  electrometer  indicated  only  a  very  small 


Fio.  441. 


Fio.  442.— (After  J.  J.  Thomson,  Conduction  of  Electricity  through  Oases.) 

effect,  but  if  the  rays  were  deflected  by  means  of  a  magnet  so  that 
they  fell  upon  the  slits  in  the  cylinders  D  and  E  the  electrometer 
indicated  that  the  cylinder  D  had  received  a  considerable  nega- 
tive charge.  If  the  rays  were  deflected  still  further  so  as  to  miss 
the  slit  the  cylinder  immediately  ceased  to  receive  any  charge. 
This  experiment  clearly  shows  that  the  rays  are  accompanied  by 
a  negative  charge  of  electricity.  If  the  cathode  rays  be  allowed 
to  pass  between  two  parallel  plates  inside  a  highly  exhausted 
cathode  ray  tube,  such  as  is  indicated  by  Fig.  442,  and  a  large 


492       CONDUCTION  OF  ELECTRICITY  THROUGH  GASES 

difference  of  potential  be  established  between  the  plates,  the 
beam  of  rays  will  be  deflected  and  the  deflection  will  be  in  the 
same  direction  as  a  negatively  charged  particle  would  be  moved 
by  the  field. 

662.  Velocity,  and  Ratio  of  the  Charge  to  the  Mass,  of  a  Cathode 
Ray  Particle.  —  We  will  now  consider  the  method  which  J.  J.  Thom- 
son originally  used  to  determine  experimentally  the  velocity  of 
these  particles  and  the  relation  between  the  mass  of  a  particle 
and  the  charge  which  it  carries.  A  highly  exhausted  cathode  ray 
tube  was  arranged  as  shown  in  Fig.  442.  C  was  the  cathode,  A 
the  anode,  and  B  a  thick  metal  plug.  A  and  B  were  pierced  by 
holes  in  the  same  straight  line  about  a  millimeter  in  diameter,  so 
that  a  very  narrow  beam  of  rays  might  pass  along  the  middle  of 
the  tube  and  fall  upon  a  screen  of  phosphorescent  material, 
thereby  producing  a  small  bright  spot.  D  and  E  were  two  parallel 
plates  which  could  be  connected  to  the  poles  of  a  battery.  Sup- 
pose that  V  is  the  velocity  of  the  particle  in  cms.  per  sec.,  m  its 
mass,  and  e  the  charge  which  it  carries,  measured  in  electro- 
magnetic units.  If  the  tube  be  placed  between  the  poles  of  a 
strong  electro-magnet,  so  that  a  field  of  strength  H  is  acting  at 
right  angles  to  the  beam,  the  spot  on  the  screen  will  move  from 
a  to  b  in  a  direction  at  right  angles  to  the  lines  of  force.  The 
cathode  particle  will  follow  a  curved  path  just  as  a  moving 
projectile  follows  a  curved  path  when  acted  on  by  gravity.  Let 
the  radius  of  curvature  of  this  path  be  r.  The  deflecting  force 
acting  along  this  radius  of  curvature  is  proportional  to  the  mag- 
netic field,  the  charge  on  the  particle,  and  its  velocity  and  is,  con- 
sequently, equal  to  HeV  (see  §433).  This  force  must  equal  the 
centrifugal  force  of  the  particle  which,  from  dynamics,  is  equal 
to  mV*/r  (see  §§32,  47).  Therefore, 


(1) 

V> 

H  can  be  measured  and  r  may  be  found  from  ab  and  the  dimen- 
sions of  the  apparatus.  Therefore  the  quantity  mV/e  is  known. 
Suppose  now  that  a  difference  of  potential  be  established  be- 
tween the  plates;  an  electric  force  will  act  on  the  beam  of  rays  and 
if  it  is  applied  in  the  right  direction  it  will  tend  to  deflect  the 


RONTGEN  RAYS  493 

beam  in  a  direction  opposite  to  the  magnetic  deflection.  Let  the 
magnetic  and  electric  forces  be  adjusted  so  that  their  effects  on 
the  particles  exactly  balance  each  other,  then  the  phosphorescent 
spot  will  return  to  the  position  it  had  before  any  force  acted  on  it. 
Let  this  electric  field  be  X.  The  force  acting  on  the  particle 
will  then  be  Xe  and,  therefore,  if  the  electric  and  magnetic  forces 
exactly  balance  each  other 

Xe  =  HeV 

(2) 


X  and  H  can  both  be  measured  and,  therefore,  V  may  be  deter- 
mined, and  knowing  V  the  value  of  e/m  is  easily  found  from 
equation  (1). 

By  this  method  Thomson  found  the  value  of  Fto  be  2.8X108 
cms.  per  second,  which  is  just  about  one  tenth  the  velocity  of 
light.  This  value  is  not  quite  constant  as  it  varies  somewhat 
with  the  potential  in  the  tube.  He  also  found  a  value  for  e/m 
the  magnitude  of  which,  according  to  later  determinations,  is 
1.7X  107;  and  he  discovered  that  it  was  independent  of  the  nature 
of  the  gas  in  the  tube. 

The  greatest  value  of  e/m  known  in  electrolysis  is  found  in  the 
case  of  the  hydrogen  ion  and  is  about  10*.  The  value  for  the 
cathode  ray  particle  is  thus  1700  times  that  for  the  hydrogen  ion. 
In  a  later  paragraph  (§564)  the  charge  e  carried  by  the  cathode 
particle  will  be  determined  and  it  may  be  shown  to  be  the  same 
as  for  the  hydrogen  ion.  Consequently,  the  mass  of  the  cathode 
particle  must  be  about  1/1700  of  the  mass  of  the  hydrogen  ion  or 
atom.  This  cathode  particle  possesses  the  smallest  mass  yet 
known,  and  is  called  an  electron  or  negative  corpuscle. 

553.  Rontgen  Rays.  —  In  1895  Rontgen  observed  that  some 
sort  of  radiation  was  produced  outside  an  ordinary  cathode  ray 
tube.  Phosphorescent  bodies  placed  near  the  tube  were  strongly 
affected  and  a  photographic  plate  in  the  neighborhood  became 
blackened.  These  radiations  have  been  called  Rontgen  rays 
after  their  discoverer.  The  name  first  applied  to  them  was  X  rays 
and  this  name  is  still  often  used. 

The  method  of  producing  Rontgen  rays  is  shown  in  Fig.  443. 
AB  is  a  large  glass  bulb.  The  cathode  a  consists  of  a  concave 
piece  of  metal,  usually  aluminum.  The  cathode  rays  proceed 


494       CONDUCTION  OF  ELECTRICITY  THROUGH  GASES 

normally  from  the  surface  of  a  and  on  account  of  its  concavity  are 
brought  to  a  focus  at  the  point  c,  and  hence  the  name  "focus  tube." 
The  anode  6  consists  in  its  simplest  form  of  a  flat  platinum  plate 
which  is  placed  at  an  angle  of  45°  to  the  axis  of  a  and  so  that  the 
center  of  curvature  of  a  is  at  the  point  c.  The  Rontgen  rays 
travel  outward  in  all  directions  from  6.  To  generate  the  rays  the 

electrodes  are  connected  to 
the  terminals  of  the  secondary 
of  an  induction  coil  or  to  an 
electrostatic  machine. 

Rontgen  rays  differ  from 
cathode  rays  inasmuch  as  they 
are  able  to  penetrate  bodies  of  considerable  thickness.  Their  pene- 
trating power,  as  well  as  some  of  their  other  properties,  depends 
upon  the  conditions  existing  within  the  tube  from  which  they  origi- 
nate. With  a  very  low  pressure  within  the  tube  and,  consequently, 
a  large  potential  difference  between  the  electrodes,  the  rays  pro- 
duced are  very  penetrating,  being  capable  of  going  through  several 
inches  of  wood  and  even  several  millimeters  of  lead.  Such  rays 
are  usually  called  "  hard  rays."  In  the  case  of  a  higher  pressure 
and  smaller  difference  of  potential  the  rays  are  less  penetrating  and 
are  called  "soft  rays."  Different  substances  absorb  the  rays  of 
any  particular  type  to  a  different  degree.  Generally  speaking 
dense  substances  produce  greater  absorption.  It  is  this  variation 
in  the  absorptive  power  of  substances  which  enables  us  to  make 
Rontgen  ray  photographs.  Rontgen  rays  act  upon  a  photo- 
graphic plate  in  a  manner  similar  to  ordinary  light  and  the  effect 
produced  depends  upon  the  intensity  of  the  rays.  Thus  a  photo- 
graph of  the  bones  of  any  portion  of  the  human  body  may  be 
obtained,  for  bones  being  denser  than  the  flesh  absorb  the  rays 
more  and,  consequently,  the  intensity  of  the  rays  which  have 
traversed  the  bones  is  less  than  the  intensity  of  those  which 
have  passed  through  only  the  flesh. 

The  Rontgen  rays  travel  in  straight  lines  with  very  high  velocity. 
Marx  has  shown  that  they  travel  with  the  velocity  of  light,  that  is, 
3  XlO10  cms.  per  sec.  No  evidence  has  as  yet  been  found  of  any 
refraction  of  the  rays  when  they  pass  from  one  medium  to 
another,  nor  has  it  so  far  been  possible  to  deflect  the  rays  by  a 
magnetic  field. 


X-RAY  SPECTROMETER  495 

It  has  been  shown  mathematically  that  when  an  electrically 
charged  particle  is  suddenly  brought  to  rest  an  electromagnetic 
disturbance  is  produced  in  the  surrounding  medium  and  travels 
outward.  This  condition  is  fulfilled  when  a  cathode  ray  particle 
is  suddenly  arrested  by  striking  against  any  solid  body.  All  of  the 
evidence  is  consistent  with  the  view  that  Rontgen  rays  are 
electromagnetic  disturbances  of  the  same  general  nature  as  light 
waves. 

554.  Reflection  of  X-rays  by  Crystals. — When  a  narrow  beam 
of  X-rays  falls  upon  a  crystalline  substance  and,  after  transmission 
by  the  crystal,  is  allowed  to  fall  upon  a  photographic  plate  some 
very  remarkable  results  are  observed.     The  photograph  obtained 
shows  a  number  of  spots  arranged  in  a  regular  manner  and  form- 
ing a  definite  pattern.     This  pattern  is  explained  by  the  principle 
of  diffraction  of  the  waves  or  pulses  of  the  X-rays  caused  by 
reflection  from  the  series  of  parallel  planes  of  which  the  crystal 
is  made  up.     Crystals  may  be  considered  as  made  up  of  atoms 
arranged  in   certain   definite  planes.     If  these  planes  form  a 
series  in  which  they  are  parallel  and  equally  spaced  then  a  pulse 
falling  upon  this  series  of  planes  will  be  reflected  from  them  and 
form  a  wave  train.     By  a  careful  study  of  these  diffraction  pat- 
terns caused  by  this  reflection  from  the  planes  of  the  crystal  a 
considerable  amount  of  information  in  regard  to  crystal  structure 
is  obtained.    For  instance  the  wave  length  of  a  homogeneous 
beam  of  X-rays  can  be  found  in  terms  of  the  dimensions  of  the 
various  parts  of  the  crystal  and  from  this  and  other  data  in  regard 
to  the  mass  of  the  atoms  certain  dimensions  of  the  crystal  can 
be  determined  and  the  wave  length  of  the  X-ray  pulse  can  be 
calculated.     By  this  means  the  various  wave  lengths  of  the  rays 
produced  by  any  X-ray  bulb  can  be  obtained  and  the  X-ray 
spectrum  determined.     By  data  of  this  kind  it  has  been  shown 
that  the  wave  length  is  characteristic  of  the  anode  in  the  bulb 
from  which  the  rays  are  produced. 

555.  X-ray  Spectrometer. — For  the  purpose  of  studying  these 
X-ray  spectra  Bragg  has  devised  a  very  ingenious  X-ray  spectro- 
meter.    It  corresponds  in  general  arrangement  to  an  ordinary 
spectrometer.     A  lead  screen  pierced  by  a  narrow  slit  through 
which  the  X-rays  pass  takes  the  place  of  the  ordinary  collimator; 
the  reflecting  crystal  occupies  the  position  of  the  usual  prism, 


496       CONDUCTION  OF  ELECTRICITY  THROUGH  GASES 

while  an  ordinary  ionization  chamber  takes  the  place  of  the  com- 
mon telescope.  The  beam  of  rays  from  the  bulb  enter  the  narrow 
slit  and  fall  upon  the  crystal  and  are  reflected  into  the  ionization 
chamber  where  the  ionization  produced  by  them  is  measured. 
The  crystal  and  the  ionization  chamber  can  be  rotated  so  that 
the  effects  for  various  angles  of  incidence  and  reflection  can  be 
measured.  By  this  means  the  reflection  produced  at  various 
angles  by  the  crystal  can  be  examined  in  detail.  This  method 
of  studying  X-rays  has  been  a  fruitful  one  in  giving  information 
in  regard  to  the  nature  of  the  rays  and  also  the  structure  of 
crystals. 

656.  Conductivity  of  Gases  Produced  by  Rontgen  Rays. — If  a 
well-insulated  body,  such  as  the  leaves  of  a  gold-leaf  electroscope 
E  (Fig.  444),  be  charged  up  in  thoroughly  dry  air  the  charge 
will  be  retained  for  many  hours.  If,  however,  a  beam  of  Rontgen 
rays  passes  through  the  gas  surrounding  the  leaves  they  will 


Fia.  444. 

immediately  lose  their  charge  and  collapse,  showing  that  the  air 
must  have  become  conducting,  allowing  the  charge  to  leak  away. 
Instead  of  the  rays  falling  directly  upon  the  gas  surrounding 
the  leaves  of  the  electroscope  let  a  system  be  arranged  as  shown 
in  Fig.  444.  AB  is  a  metal  tube  through  which  a  stream  of  air 
may  be  sent  and  which  leads  into  an  electroscope  E.  If  the  Ront- 
gen rays  fall  upon  the  air  in  the  part  AC,  no  effect  is  produced  in 
the  electroscope  as  long  as  there  is  no  stream  passing  through  the 
tube;  but  as  soon  as  a  stream  of  air  is  passed  through  the  tube 
into  E  the  leaves  lose  their  charge.  This  conducting  property 
imparted  to  the  air  by  the  rays,  therefore,  may  be  transported 
along  with  the  air.  If  a  plug  of  cotton  wool  be  placed  in  the  tube 
at  C,  or  if  the  air  be  bubbled  through  water,  after  being  acted 


GASES  PRODUCED  BY  RONTGEN  RAYS  497 

upon  by  the  rays  this  conductivity  is  entirely  destroyed.  If  an 
insulated  wire  ab  be  introduced  in  the  center  of  the  tube  CB  and 
a  strong  electric  field  be  established  between  the  wire  and  the 
tube,  by  connecting  the  wire  to  one  pole  of  a  battery  and  the  tube 
to  the  other  pole,  the  air  loses  its  conductivity  in  passing  through 
the  tube. 

The  removal  of  this  conducting  power  from  the  gas  by  filtering 
it  through  cotton  wool  or  water  indicates  that  the  conductivity 
must  be  due  to  something  mixed  with  the  air,  while  its  removal 
by  an  electric  field  shows  that,  whatever  it  may  be  that  is  mixed 
with  the  air,  it  must  carry  an  electric  charge. 

Suppose  again  that  A  and  B,  Fig.  445,  are  two  parallel  metal 
plates  placed  a  few  centimeters  apart  A  B 

in  air  and  let  A  be  connected  to  one  C2 
pole  of  a  battery  while  the  other  pole  /^ 
is  connected  to  earth;  let  B  be  con-  '  Earth 

nected  to  one  pair  of  quadrants  of  a  ^^^^^-^^ 

quadrant  electrometer  while  the  other  ff 

pair   of    quadrants   is  connected    to 

earth.  If  a  beam  of  Rontgen  rays  be  passed  between  these 
plates  it  will  be  observed  that  B  immediately  begins  to  receive 
a  charge,  as  indicated  by  the  deflection  of  the  electrometer  needle. 
It  will  continue  to  charge  up  as  long  as  the  rays  are  acting,  but 
will  cease  if  the  rays  cease.  If  Cj  is  the  positive  pole  of  the 
battery  then  B  will  receive  a  positive  charge,  but  if  the  poles  be 
reversed  B  will  receive  a  negative  charge.  The  rays  thus  ap- 
parently cause  a  transference  of  electricity  through  the  air  to  B 

and  the  sign  of  the  electric  charge 
given  to  B  depends  upon  the    sign 
-*    Of  A. 

*    / 

657.  Saturation  Current. — If  the  poten- 
tial difference  between  A  and  B  be  altered 
_  the  charge  received  by  B  in  a  given  time 

E.M»F.  alters,  that  is,  the  current  between  A  and 

FIG.  446.  B  depends  upon  the  voltage.     The  current 

through  the  gas  does  not,  however,  obey 

Ohm's  law,  for  if  the  current  corresponding  to  different  voltages  be 
measured  and  a  curve  plotted  showing  the  relation  between  current 
and  voltage,  it  will  assume  the  form  shown  in  Fig.  446  instead  jof  being 
a  straight  line.  It  will  be  seen  that  for  small  voltages  _the  current 
32 


498       CONDUCTION  OF  ELECTRICITY  THROUGH  GASES 

obeys  Ohm's  Law,  but  it  soon  begins  to  fall  off  and  finally  reaches 
a  constant  value  even  for  a  large  increase  in  voltage.  This  character- 
istic curve  has  been  called  a  saturation  curve  on  account  of  its  similarity 
in  form  to  the  saturation  curve  in  the  magnetization  of  iron.  The 
current  corresponding  to  the  flat  part  of  the  curve  is  called  the  saturation 
current. 

The  current  through  a  gas  differs  very  markedly  in  another  respect  from 
the  current  through  metals  or  liquids.  When  the  distance  between  two 
electrodes  immersed  in  a  liquid  is  increased  the  current  decreases  on  account 
of  the  increase  of  resistance  between  the  electrodes,  but  in  the  case  of  a 
gas  the  saturation  current  increases  when  the  distance  between  the  plates 
is  increased.  Within  certain  distances  the  saturation  current  is  propor- 
tional to  the  distance  between  the  plates. 

558.  Theory  of  lonization  of  Gases. — These  facts  along  with 
others  have  led  to  the  ionization  theory  of  gases.  According  to 
this  theory  Rontgen  rays,  when  they  pass  through  a  gas,  cause  the 
molecules  of  the  gas  to  be  broken  up  into  positively  and  nega- 
tively charged  carriers  of  electricity  called  ions.  This  process 
of  breaking  up  the  molecule  is  called  ionization  and  the  gas  is 
said  to  be  ionized.  From  each  molecule  ionized  two  ions,  having 
equal  charges  but  of  opposite  sign,  are  produced.  The  trans- 
ference of  electricity  through  the  gas  is  due  to  the  movement  of 
these  charged  carriers  under  the  influence  of  an  electric  field. 
The  positive  ions  are  attracted  to  the  negative  electrode  and 
the  negative  ions  to  the  positive  electrode,  and  the  movement 
of  these  electric  charges  constitutes  a  current.  When  the  gas 
is  passed  through  the  tube  with  a  central  wire,  between  which 
there  is  an  electric  field,  the  positive  and  negative  ions  are 
attracted  to  the  negative  and  positive  electrodes  respectively  and 
thus  removed.  When  the  gas  is  passed  through  cotton  wool  the 
ions  are  caught  by  the  wool. 

659.  Explanation  of  Saturation  Current. — The  above  theory  also  explains 
the  saturation  curve  for  a  current  between  two  plates.  The  current  is  pro- 
portional to  the  number  of  ions  reaching  the  plates  per  second  and,  there- 
fore, to  the  potential  difference  provided  this  is  not  too  high.  But  when  the 
voltage  reaches  a  certain  value  the  ions  move  so  fast  that  they  practically  all 
reach  the  plates  before  they  have  time  to  recombine,  and  the  current  could 
not  be  increased  further  even  by  a  higher  voltage,  as  the  number  of  ions 
removed  could  not  be  augmented. 

The  increase  of  current  between  two  plates  when  the  distance  between 
them  is  lengthened  is  also  easily  explained  by  this  theory.  When  the  plates 
are  placed  farther  apart  the  volume  of  gas  acted  on  by  the  rays  is  increased 


IONIZATION  BY  COLLISION  499 

and,  consequently,  the  number  of  ions  produced  grows  greater  in  the  same 
proportion,  and  a  greater  number  will  reach  the  plates  per  second  and  the 
maximum  current  be  raised. 

660.  Effect  of  Conditions  on  lonization. — The  nature  and  quality  of  the 
ionizing  rays  determine  the  number  of  ions  produced  in  any  given  gas. 
Cathode  and  Rflntgen  rays,  for  instance,  differ  in  ionizing  power  and  even 
R6ntgen  rays  differ  among  themselves  in  this  respect.  Penetrating  rays  of 
any  type  are  usually  less  powerful  ionizers  than  those  less  penetrating.  For 
a  constant  ionizing  source  the  number  of  ions  produced  in  a  given  volume 
of  gas  is  found  experimentally  to  be  directly  proportional  to  the  pressure. 
Temperature  on  the  other  hand  has,  as  far  as  is  known,  no  effect  on  ioniza- 
tion,  if  the  density  of  the  gas  is  kept  constant. 

561.  Recombination  of  Ions. — When  the  rays  begin  to  ionize  the  gas  the 
ions  gradually  increase  in  number  until  a  steady  state  is  reached,  when  no 
further  increase  will  take  place  no  matter  how  long  the  rays  act.     As  the 
rays  are  continually  producing  ions  they  must  be  disappearing  at  the  same 
rate  as  they  are  being  produced  when  this  steady  state  is  reached.     Being 
positively  and  negatively  charged  bodies  and  being  in  motion  they  collide 
and  neutralize  each  other  electrically  and  disappear  as  far  as  producing  any 
conductivity  is  concerned. 

562.  Diffusion  of  Ions. — The  ions  of  an  ionized  gas  are  in  motion  and  if 
there  is  an  excess  of  ions  in  one  part  of  the  gas  they  will  diffuse  to  the  other 
part.     If  the  ionized  gas  is  in  an  enclosed  vessel  the  ions  will  diffuse  to  the 
sides  of  the  vessel  and  disappear  from  the  gas.     Sometimes,  in  a  very  con- 
fined space,  the  loss  of  ions  by  diffusion  is  even  more  important  than  the  loss 
by  recombination. 

A  detailed  study  of  diffusion  has  led  to  the  theory  that  both  the  positive 
and  negative  ions,  at  ordinary  pressures,  consist  of  a  cluster  of  molecules 
surrounding  a  charged  nucleus.  lonization  is  considered  to  consist  in  sepa- 
rating a  negative  electron  from  the  neutral  molecule  and  then  the  electron 
becomes  loaded  with  a  cluster  of  molecules  and  forms  the  negative  ion  under 
ordinary  conditions.  The  positive  ion  consists  to  begin  with  of  the  molecule 
deprived  of  the  electron  and  then  a  cluster  of  molecules  is  formed  about  this 
positively  charged  center.  The  positive  and  negative  ions  diffuse  more 
nearly  at  the  same  rate  in  moist  than  in  dry  gases,  for  in  dry  gases  the  nega- 
tive ion  is  smaller,  but  in  a  moist  gas  it  becomes  more  loaded  up  with  moisture 
than  the  positive  ion  and  its  rate  of  diffusion  decreases  more  rapidly.  As  the 
pressure  of  the  gas  is  lowered  the  coefficient  of  diffusion  of  the  negative  ion 
increases  faster  than  that  of  the  positive  and  it  has  been  shown  that  at  low 
pressures  the  negative  ion  is  the  same  as  the  electron. 

663.  lonization  by  Collision. — In  §557  the  current-voltage 
curve  for  a  gas  at  atmospheric  pressure  showed  a  final  maximum 
current.  In  a  gas  at  low  pressures,  in  the  neighborhood  of  1  mm. 
of  mercury  a  new  phenomenon  appears  and  the  corresponding 
curve  for  current  and  voltage  assumes  the  from  shown  in  Fig. 


500       CONDUCTION  OF  ELECTRICITY  THROUGH  GASES 

447.  For  low  voltages  the  part  of  the  curve  up  to  a  point  A  is 
of  the  same  form  as  the  saturation  curve  at  atmospheric  pressure, 
but  when  the  voltage  is  increased  beyond  a  certain  amount  the 
current  begins  to  increase  again,  at  first  slowly  and  then  very 
rapidly.  The  increase  of  current  beyond  the  point  A  must  be 
caused  by  an  increase  in  the  number  of  ions  due  to  some  cause 
other  than  the  original  ionizing  agency.  This  larger  number  of 
ions  has  been  explained  by  the  theory  that  if  an  ion  is  moving 
with  sufficient  velocity  it  will  produce  more  ions  by  collision  with 
the  molecules  of  the  gas.  A  moving  ion 
possesses  kinetic  energy  and  if  its  velocity 
is  great  enough  it  will  possess  sufficient 
energy  to  ionize  a  molecule  with  which  it 
may  collide.  The  kinetic  energy  depends 
upon  the  velocity  and  this,  in  turn,  depends 
upon  the  electric  field  and  upon  the  oppor- 
—  tunity  the  ion  has  of  acquiring  speed  among 


FIG  447  ^e  m°lecules  °f  the  gas.  At  atmospheric 

pressure  the  molecules  are  so  close  together 
that  the  ion  is  not  able,  between  two  collisions,  to  acquire  suf- 
ficient velocity  in  ordinary  electric  fields  to  ionize  a  molecule, 
but  at  low  pressures  the  molecules  are  so  few  in  number  and  so 
far  apart  that  the  ion  may  acquire  sufficient  speed  between 
collisions  to  ionize  any  molecule  which  it  strikes.  This  produc- 
tion of  ions  by  collision  is  only  observed  for  ordinary  electric  fields 
at  pressures  below  about  30  mm. 

The  above  theory  of  ionization  by  collision  furnishes  a  very 
satisfactory  explanation  of  the  electric  spark  through  a  gas  at 
atmospheric  pressure.  There  are  always  in  gases  a  few  ions 
which  may  be  detected  by  the  use  of  sensitive  instruments.  If  a 
voltage  high  enough  to  produce  a  spark  is  established  between  two 
points,  the  few  ions  naturally  present  in  the  field  will  acquire  a 
velocity  sufficient  to  ionize  any  molecules  against  which  they 
strike;  these  new  ions  will  in  turn  produce  more  ions,  and  so  the 
number  will  increase  very  rapidly  until  there  are  enough  to  carry 
a  current,  and  this  current  is  the  electric  spark. 

564.  Charge  carried  by  an  Ion. — It  was  known  for  some  years 
that  if  dust  particles  were  present  in  a  damp  gas  the  water  vapor 
would  condense  around  these  nuclei  when  a  sudden  expansion  of 


EMISSION  OF  ELECTRONS  BY  METALS  501 

the  gas  took  place.  If  a  beam  of  X  rays  is  allowed  to  fall  upon 
a  steam  jet,  condensation  takes  place,  the  ions  produced  acting 
as  nuclei  on  which  water  vapor  condenses. 

This  property  of  ions  to  act  as  condensation  nuclei  has  been 
utilized  by  J.  J.  Thomson  to  determine  the  absolute  value  of  the 
charge  carried  by  an  ion.  When  an  expansion  takes  place  in 
ionized  air  water  drops  form  around  the  ions  and  fall  under  the 
action  of  gravity.  Sir  George  Stokes  has  shown  that  a  drop  of 
water  of  radius,  r,  falls  through  a  gas  of  viscosity,  //,  with  the 
velocity,  v,  given  by  the  equation 

2gr* 

"=9y 

where  g  is  the  acceleration  of  gravity.  The  velocity,  v,  can  be 
measured  by  observing  the  rate  at  which  the  cloud  falls  under  the 
action  of  gravity,  and  since  g  and  /*  are  known  r  may  be  deter- 
mined. If  m  is  the  mass  of  water  deposited  and  n  the  number  of 

drops  per  c.c.  then  w  =  nX^r8,  since  the  density  of  water  is 

o 

unity.  The  amount  of  water  vapor  deposited  when  a  known  ex- 
pansion occurs  can  be  easily  calculated  from  well-known  thermal 
considerations  and,  therefore,  m  may  be  determined.  Knowing 
m  and  r  the  number  of  drops,  n,  which  is  the  same  as  the  number 
of  ions,  is  easily  calculated. 

If  all  the  ions  present  be  extracted  by  an  electric  field  between 
two  electrodes  in  the  usual  way,  the  total  charge  carried  by  all  the 
ions  can  be  measured.  Knowing,  therefore,  the  number  of  ions 
and  the  total  charge  on  them,  the  charge  carried  by  each  one  is 
determined.  By  a  modification  of  this  method  using  a  single 
drop  instead  of  the  cloud  Millikan  has  shown  this  charge  to  be 
equal  to  4.774xlO~10  electrostatic  units.  It  has  been  shown 
that  the  charge  acquired  by  such  a  drop  suspended  in  space  is 
always  an  exact  multiple  of  the  elementary  charge.  The  charge 
carried  by  ions  in  hydrogen  and  oxygen  has  the  same  value  and 
does  not  depend  upon  the  source  from  which  they  are  produced. 

565.  Emission  of  Electrons  by  Metals. — If  ultra-violet  light  rays 
fall  upon  the  clean  suf ace  of  a  piece  of  zinc,  sodium,  potassium, 
lithium,  etc.,  which  is  negatively  charged  the  metal  will  lose  its 
charge,  while  if  the  metal  be  uncharged  to  begin  with  it  will  ac- 
quire a  positive  charge.  If  the  metal  is  positively  charged  to 


502  RADIO-ACTIVITY 

begin  with  no  loss  of  charge  takes  place.  These  photo-electric 
effects  as  they  are  called,  have  been  shown  to  be  due  to  the 
liberation  of  negative  corpuscles,  or  electrons,  from  the  metal 
by  the  action  of  the  ultra-violet  light. 

If  a  metal  electrode  be  placed  near  to  a  metal  wire  and  the 
latter  be  then  heated  until  it  begins  to  glow,  a  current  through 
the  gas  will  be  produced  and  the  electrode  will  receive  a  charge. 
A  platinum  wire  heated  to  redness  will  under  some  conditions 
give  a  positive  charge  to  the  other  electrode,  but  if  heated  to  white 
heat  the  charge  is  negative.  The  behavior  of  hot  metals  is 
somewhat  irregular  but  in  general  metals  and  carbon  heated  to 
incandescence  in  high  vacua  give  off  negatively  charged  carriers. 
The  ratio  of  the  charge  to  the  mass  of  these  carriers  has  been  shown 
to  be  the  same  as  for  the  cathode  ray  particles  and  the  electron 
liberated  by  ultra-violet  light  at  low  pressures.  This  along  with 
other  considerations  has  led  to  the  theory  that  these  negative 
corpuscles  are  distributed  throughout  the  volume  of  metals  at  all 
temperatures,  but  when  the  metals  are  heated  to  incandescence 
the  corpuscles  then  acquire  sufficient  energy  to  escape  into  the 
the  surrounding  space. 

566.  lonization  by  Flames. — If  two  electrodes  are  placed  some  distance 
apart  in  an  ordinary  Bunsen  flame  quite  an  appreciable  current  is  observed 
which  may  be  measured  by  a  galvanometer.  If  the  air  surrounding  such  a 
flame  be  drawn  away  from  the  flame  it  is  found  to  be  still  a  conductor.  The 
ions  which  have  been  produced  in  the  gas  by  the  flame  appear  to  be  much 
larger  than  those  produced  in  other  ways,  for  their  velocity  has  been  meas- 
ured and  found  to  be  much  less  than  that  of  other  ions.  It  is  due  to  this 
conducting  power  of  flames  that  when  an  insulator  has  received  an  electro- 
static charge  it  may  be  discharged  by  simply  passing  a  Bunsen  flame  over  it 

RADIO-ACTIVITY 

667.  Discovery  of  Radio -Activity. — The  phosphorescent  action 
of  Rontgen  rays  led  physicists  to  investigate  phosphorescent  sub- 
stances and  Becquerel  in  1896  found  that  the  double  sulphate  of 
uranium  and  potassium  emitted  a  radiation  which  produced  an 
effect  upon  a  photographic  plate  similar  to  that  of  X-rays.  He 
later  examined  other  compounds  of  uranium  as  well  as  the  ele- 
ment itself  and  found  that  they  all  possessed  this  power.  Al- 
though the  phosphorescent  action  of  RGntgen  rays  pointed  the 


RADIO-ACTIVE  SUBSTANCE  503 

way  to  this  discovery,  it  has  since  been  shown  that  there  is  no 
connection  between  the  rays  emitted  by  uranium  and  its  phos- 
phorescence, for  some  compounds  which  are  not  phosphorescent 
emit  the  rays. 

Becquerel  and  others  showed  that  these  radiations  from 
uranium  were  capable  of  discharging  electrified  bodies  and  that 
this  power  of  discharging  electrified  bodies  was  due  to  the  produc- 
tion by  these  rays  of  ions  in  the  gas,  similar  to  the  ions  produced 
by  Rontgen  rays. 

If  the  rays  from  uranium  be  allowed  to  pass  between  two  parallel 
plates,  between  which  there  is  a  difference  of  potential,  a  current 
will  pass  through  the  air  just  as  in  the  case  of  a  gas  ionized  by 
Rontgen  rays.  Thus  suppose  that  A  and  B  (Fig.  448)  are  two 
insulated  metal  plates.  The  upper  plate  A  is  connected  to  one 
pair  of  quadrants  of  an  electrometer,  the  other  pair  being  to 
earth.  If  a  layer  of  one  of  the  compounds  of  uranium  be  sprinkled 
on  the  plate  B,  as  indicated, 
an  ionization  current  will  be  C 

produced  between  A  and  B.     A 

This  property  of  uranium      n  Earth 

does  not  deteriorate  with 
time.  Uranium  and  other 
bodies,  which  possess  similar 
properties,  are  called  radio-  FIQ  44g 

active  bodies. 

568.  Other  Radio-active  Substances. — Schmidt,  and  inde- 
pendently Mme.  Curie,  discovered  that  the  element  thorium  and 
its  compounds  possess  radio-active  properties.  The  photo- 
graphic action  of  thorium  was  found  to  be  distinctly  weaker  than 
that  of  uranium,  while  the  ionizing  action  was  about  equal  to 
that  of  uranium,  but  was  very  irregular.  A  very  systematic 
examination  of  a  large  number  of  minerals  containing  uranium 
and  thorium  was  then  undertaken.  Using  the  electrical  method 
the  current  produced  between  two  plates  by  a  given  amount  of 
each  of  the  minerals  was  measured.  The  results  showed  that  all 
minerals  containing  uranium  or  thorium  were  radio-active,  but 
that  several  specimens  of  pitchblende,  as  well  as  some  other 
minerals,  were  several  times  more  active  than  uranium  itself. 
T  is  led  to  the  conclusion  that  there  must  be  some  other  and 


504  RADIO-ACTIVITY 

more  active  substance  in  pitchblende.  M.  and  Mme.  Curie  then 
investigated  this  question  chemically  and  discovered  two  new 
active  bodies. 

The  first  of  these  substances  to  be  separated  by  purely  chem- 
ical means  was  found  to  be  very  much  more  active  than  uranium 
and  it  was  given  the  name  polonium  in  honor  of  Mme.  Curie's 
native  country.  Polonium  differs  from  uranium  in  the  essential 
particular  that  its  activity  is  not  constant  but  gradually  dies 
away.  In  some  cases  it  was  found  that  at  the  end  of  about  six 
months  after  preparation  the  activity  had  fallen  to  half  its  origi- 
nal value. 

The  other  active  substance  discovered  in  pitchblende  was 
found  to  be  enormously  more  active  than  uranium.  In  its  pure 
state  it  is  about  a  million  times  more  active,  and  it  was  called 
radium  by  the  discoverers.  The  quantity  of  radium  in  pitch- 
blende is  almost  infinitesimal,  about  a  ton  of  pitchblende  con- 
taining only  a  few  milligrams  of  pure  radium.  Radium  is  found 
in  varying  quantities  in  a  number  of  minerals  and  in  various 
parts  of  the  world.  For  some  years  the  pitchblende  found  in 
Bohemia  furnished  most  of  the  radium,  but  quite  recently  con- 
siderable quantities  have  been  obtained  from  the  carnotite  ores 
found  in  Colorado. 

In  practice  radium  is  not  separated  from  its  compound  but  is 
usually  employed  in  the  form  of  the  bromide,  and  what  is  often 
called  "pure  radium "  is  really  "pure  radium  bromide."  It  also 
forms  other  compounds,  such  as  the  chloride,  sulphate,  etc., 
and  these  salts  are  all  naturally  phosphorescent  and  their 
radiations  produce  phosphorescence  in  various  substances  such 
as  platino-barium  cyanide,  willemite,  etc. 

Debierne,  in  analyzing  residues  from  pitchblende,  discovered 
a  very  active  substance  which  he  called  actinium.  The  properties 
of  actinium  are  very  similar  to  those  of  thorium,  but  the  former 
is  very  many  times  more  active  than  the  latter.  Actinium 
besides  being  strongly  radio-active  is  capable,  like  radium,  of 
producing  phosphorescence  in  such  substances  as  zinc  sulphide, 
willemite,  etc. 

669.  Three  Types  of  Rays. — In  examining  the  radiations  from 
uranium  Rutherford  found  that  there  were  two  distinct  types  of 
rays,  one  type  which  were  easily  absorbed  by  solid  bodies,  and  a 


NATURE  OF  THE  MASS  OF  AN  ELECTRON  505 

second  type  which  were  more  penetrating  and,  besides,  could  be 
easily  deflected  from  their  path  by  a  magnetic  field.  The  former 
he  called  a  rays  and  the  latter  ft  rays.  Later  it  was  shown  that 
there  was  still  a  third  type  emitted  which  were  extremely  pene- 
trating and  could  not  be  deflected  by  a  magnetic  field.  These 
were  called  7-  rays.  The  four  radio-active  substances  uranium, 
thorium,  radium  and  actinium,  under  normal  conditions,  give 
out  these  three  types  of  rays.  Polonium,  however,  emits  only 
a  rays. 

670.  The  ft  Rays. — Becquerel,  using  the  photographic  method, 
showed  that  the  ft  rays  of  radium  behaved  in  every  respect  like 
cathode  rays.     They,  consequently,  must  be  negatively  charged 
particles  or  electrons. 

Combining  deflections,  by  a  magnetic  and  by  an  electric 
field,  in  a  manner  somewhat  similar  in  principle  to  that  used 
in  the  case  of  the  cathode  rays  (§552),  Becquerel  determined  the 
velocity  and  the  ratio  of  e/m  for  the  ft  rays.  For  e/-m  he  found  a 
value  which  does  not  differ  much  from  the  value  found  for  the 
cathode  rays  or  electrons.  He  observed,  however,  that  the  ft 
rays  did  not  all  have  the  same  velocity  as  some  were  bent  more 
than  others.  He  showed  that  the  velocities  varied  from  about 
6X109  to  2.8X1010  cms.  per  sec.,  the  latter  approaching  very 
nearly  the  velocity  of  light  which  is  3  X 1010  cms.  per  sec.  The 
ft  rays  from  radium  appear,  therefore,  to  be  complex,  being  a 
mixture  of  rays  of  the  same  nature  but  travelling  with  different 
speeds.  The  ft  rays  from  uranium  differ  from  those  of  radium  in 
this  respect  for  the  former  appear  to  be  homogeneous. 

671.  Nature  of  the  mass  of  an  electron. — This  complexity  of 
the  ft  rays  (or  electrons)  with  regard  to  velocity  led  Kauf mann  to 
examine  whether  the  value  of  elm  for  these  rays  varied  with  the 
speed.     He  showed  that  e/m  decreased  when  the  speed  increased. 
Assuming  that  the  charge  on  the  ft  ray  particle  is  constant  the 
mass  of  the  particle  appears  to  increase  with  increase  of  velocity. 

Several  mathematical  physicists  have  worked  out  from  purely 
theoretical  considerations  that  the  apparent  mass  of  a  moving 
electron  is  due,  either  wholly  or  in  part,  to  the  electric  charge  in 
motion,  that  is,  when  an  electric  charge  is  moving  it  appears  to 
possess  what  corresponds  to  inertia,  due  to  the  fact  of  its  being 
in  motion.  This  apparent  inertia  according  to  this  view  is  not  due 


506 


RADIO-ACTIVITY 


to  material  mass  as  we  are  accustomed  to  conceive  of  it,  but  is  a 
result  of  the  motion  of  the  electric  charge.  These  theoretical 
considerations  further  show  that  this  apparent  mass,  which 
seems  to  be  electrical  in  origin,  increases  with  the  speed  of  the 
moving  charge.  Experimental  results  seem  to  confirm  the 
theoretical  view  that  the  mass  of  the  electron  is  due,  wholly  or  in 
part,  to  the  fact  that  the  electric  charge  is  in  motion. 

572.  The  a  Rays. — The  first  attempts  to  deflect  a  rays  by  a  mag- 
netic field  and  so  ascertain  their  nature  failed.  Rutherford 
succeeded  in  doing  this  by  using  intense  radiation  and  a  very 
powerful  field.  His  apparatus  is  shown  diagrammatically  in 
Fig.  449. 

A  is  a  gold-leaf  electroscope,  SS,  a  set  of  parallel  brass  plates 
separated  by  very  narrow  slits  the  width  of  which  was  in  some 
experiments  as  small  as  0.042  cm.  but  varied  for  different  ex- 
periments up  to  0.1  cm.  A  quantity  of  radium,  R,  was  placed 
below  the  slits  and  the  rays  passed  up  through  them  and  into 
the  electroscope  where  they  ionized  the  air.  Of  course,  the  /? 
and  7-  rays  were  also  present  but  the  ionization  produced  by  the  a 
rays  was  more  than  nine  times  that  produced  by  the  /?  and  f 
rays  combined,  so  their  presence  did  not  affect  the  experiment. 
By  applying  a  magnetic  field  in  a  direction  parallel  to  the  slits 
and  at  right  angles  to  the  plane  of  the  paper  the  rays,  if  they  are 
deviable,  should  be  bent  either  to  the  right  or  left  and  strike  the 

plates  and  be  stopped  before  they  could 
4g=— Hydrogen  emerge  beyond  the  slits.  He  found  that 
by  the  application  of  the  magnetic  field 
over  eight-ninths  of  the  a  radiation  could 
be  cut  off,  showing  that  the  a  rays  could 
be  deviated  by  the  field.  By  a  slight 
modification  of  the  experiment  he  showed 
that  they  were  bent  in  the  opposite  direc- 
tion to  that  in  which  the  /?  rays  would  be 
bent,  indicating  that  the  a  rays  must 
carry  a  positive  charge.  He  also  succeeded  in  deflecting  the  a 
rays  by  an  electric  field  using  an  apparatus  similar  to  the  one 
just  described. 

Experiments  show  that  within  the  limits  of  experimental  error 
the  value  of  e/m  is  the  same  for  the  «  rays  emitted  by  the  various 


Earth 

Hydrogen 
FIG.  449. 


THE  a  RAYS  507 

radio-active  substances.  The  average  experimental  value  ob- 
tained is  about  5X103  electromagnetic  units.  Assuming  that 
the  charge  on  each  particle  is  the  same,  the  mass  of  the  a 
particles  emitted  by  the  different  substances  is  constant. 

Although  the  mass  is  constant  yet  the  velocity  of  expulsion  of 
the  a  particles  is  not  the  same  for  all  substances,  as  it  is  found  to 
vary  from  1 . 56  X 109  to  2 . 25  X 109  cms.  per  second. 

573.  Mass  and  Nature  of  a  Particle. — These  results  enable  us  to 
obtain  a  more  definite  idea  of  the  mass  and  nature  of  the  a  particle. 
The  value  of  e/m  for  the  atom  of  hydrogen  liberated  in  the 
electrolysis  of  water  is  about  104  electromagnetic  units,  while  we 
have  just  seen  that  for  the  a  particle  e/m  is  5  X  10s.  Rutherford 
showed  that  within  the  limits  of  experimental  error  the  charge 
carried  by  the  a  particle  is  twice  the  charge  carried  by  a  gaseous 
ion  and  consequently  twice  the  charge  on  the  electrolytic  hydro- 
gen ion  or  atom.  It  follows  from  this  that  the  mass  of  the  a 
particle  must  be  four  times  the  mass  of  the  hydrogen  atom. 
Since  it  is  atomic  in  size  and  of  the  same  order  as  the  atom  of 
helium  (whose  atomic  mass  is  3.96  in  terms  of  hydrogen)  and 
since  there  does  not  seem  to  be  any  place  according  to  the  periodic 
law  among  the  elements  for  a  new  one  in  that  part  of  the  series 
the  most  natural  hypothesis  is  that  the  a  particle  is  an  atom  of 
helium  carrying  twice  the  ionic  charge  of  hydrogen.  Helium  is 
continually  produced  by  both  radium  and  actinium.  As  a  final 
proof  it  has  been  shown  that  when  a  particles  are  allowed  to 
penetrate  into  a  vacuum  helium  always  accumulates. 

674.  Absorption  of  a  Rays. — A  distinguishing  characteristic  of  the  a 
rays  is  that  they  are  very  easily  absorbed  when  passing  through  either 
gases  or  solids.  The  proportion  of  the  rays  absorbed  by  a  given  thickness 
of  any  solid  may  be  determined  by  first  measuring  the  saturation  current 
produced  by  the  rays,  and  then  covering  the  radiating  material  with  the 
absorbing  solid  and  again  measuring  the  current  produced  by  the  rays 
after  they  have  passed  through  the  solid.  The  absorbing  layer  must  be 
very  thin  or  else  all  the  rays  will  be  stopped.  The  most  penetrating  a  rays 
known  are  completely  absorbed  by  a  thickness  of  only  about  0.006  cm. 
of  aluminum.  The  penetrating  power  of  the  a  rays  varies  greatly  with  the 
different  substances  from  which  they  are  emitted. 

The  a  rays  are  very  easily  stopped  by  gases,  a  few  centimeters  of  air 
at  atmospheric  pressure  being  sufficient  to  absorb  them,  consequently,  the 
ionization  produced  by  them  exists  only  within  a  few  centimeters  of  the 
source  from  which  the  rays  come.  The  absorption  by  gases  depends  upon 


508  RADIO-ACTIVITY 

the  density,  being  in  some  cases  proportional  thereto,  but  not  so  in  all 
cases.  The  absorption  of  the  rays  by  gases  is  important  as  the  degree  of 
ionization  produced  by  the  rays  depends  upon  the  amount  of  the  rays 
absorbed,  the  relative  ionization  by  the  a  rays  in  gases  being  directly  pro- 
portional to  the  relative  absorption. 

676.  The  y  Rays. — The  third  distinct  type  of  rays  given  out  by 
some  of  the  radio-active  substances  differs  very  essentially  from 
the  a  and  /?  rays.  The  y  rays  are  extremely  penetrating,  being 
capable  of  passing  through  large  thicknesses  of  solid  matter. 
For  instance,  the  y  rays  given  out  by  very  strong  radium  bromide 
can  be  detected  after  passing  through  30  cms.  of  iron.  They  are 
very  much  more  penetrating  than  the  X  rays  from  a  very  hard 
X  ray  bulb.  They  ionize  gases  but  to  a  very  much  less  extent 
than  either  the  a  or  ft  rays,  the  ionization  being  very  approxi- 
mately proportional  to  the  density  of  the  gas. 

No  one  has  as  yet  succeeded  in  deviating  the  y  rays  by  either  an 
electric  or  magnetic  field.  Their  great  penetrating  power  and 
their  non-deviability  show  a  close  resemblance  to  very  hard 
X  rays.  We  know  also  that  X  rays  are  produced  by  the  sudden 
stopping  of  a  moving  electron,  and  it  is  reasonable  to  suppose 
that  they  would  be  produced  by  the  sudden  starting  of  an  electron. 
Now  experiment  has  shown  that  y  rays  occur  only  in  conjunction 
with  ft  rays  and  the  /?  rays  we  know  are  electrons.  Consequently, 
it  is  reasonable  to  suppose  that  the  y  rays  are,  like  X  rays,  electro- 
magnetic pulses  produced  by  the  sudden  emission  of  the  /?  particle, 
or  electron  from  the  radio-active  substance.  This  theory  seems 
to  be  supported  strongly  by  the  evidence  at  present  available, 
although  it  is  very  difficult  to  settle  the  question  definitely  by 
direct  proof. 

576.  Production  of  Uranium  X  and  Thorium  X. — Crookes  in 
1900  showed  that  by  a  simple  chemical  process  he  could  separate 
from  uranium  a  constituent  which  was  many  times  more  active 
photographically  than  the  uranium  from  which  it  was  separated 
and,  in  addition,  the  separation  of  this  constituent  left  the  ura- 
nium photographically  inactive.  This  new  and  unknown  con- 
stituent he  called  Uranium  X,  or  Ur.  X.  Becquerel  obtained 
similar  results  using  a  slightly  different  chemical  process,  and,  on 
testing  about  a  year  later  the  Ur.  X  and  the  uranium  from  which 
it  had  been  separated,  discovered  in  addition  the  curious  fact 


URANIUM  X  AND  THORIUM  X 


509 


that  the  uranium  had  completely  recovered  its  usual  amount  of 
activity  while  the  Ur.  X  had  entirely  lost  its  activity.  Ruther- 
ford and  Soddy  later  succeeded  in  performing  a  similar  chemical 
operation  with  thorium,  separating  a  very  active  constituent  from 
thorium,  which  they  called  Thorium  X  or  Th.  X  and  which  acted 
in  a  manner  very  similar  to  Ur.  X. 

These  phenomena  have  been  thoroughly  examined,  both  by 
the  photographic  and  electrical  methods,  and  it  has  been  found 
in  the  case  of  uranium  that  after  separation  the  Ur.  X  was  very 
active  photographically  but  inactive  electrically,  because  it  gave 
out  /?  rays  but  no  a  rays,  while  the  uranium  from  which  it  had 
been  separated  was  inactive  photographically  but  still  active 
electrically,  owing  to  the  fact  that  it  gave  out  a  rays  but  practic- 
ally no  /?  rays.  The  Ur.  X  gradually  lost  its  activity,  while  the 
uranium  regained  its  /?  ray  activity  again,  and  the  loss  in  the 
one  instance  and  the  recovery  in  the  other  took  place  at  the  same 
rate.  When  the  Ur.  X  had  lost  half  its  activity  the  uranium  had 
regained  half  its  original  activity  and  each  process  took  about 
20.7  days.  The  way  in  which  this  occurred  is  shown  very  clearly 
by  the  curves  in  Fig.  450  which  represent  the  activity  of  each  at 
different  times  after  separation,  the  ordinates  representing 
activity  and  the  abscissae 
time  in  days.  Similar  re-  "|100 
suits  but  of  a  slightly  more 
complicated  nature  have 
been  observed  for  thorium, 
but  the  time  taken  for  the 
activity  of  Th.  X  to  decay 
to  half  its  maximum  value 
and  that  of  the  thorium  to 
regain  half  its  activity  have 
been  found  to  be  only  6.64  days. 

These  results  indicate  that  some  process  must  be  continually 
going  on  in  these  substances.  Since  the  Ur.X  which  gives  out  /? 
rays  can  be  separated  from  the  normal  uranium  leaving  it  devoid 
of  /?  rays,  therefore,  the  /?  rays  must  arise  from  the  Ur.  X,  and 
since  the  uranium  regains  the  /?  ray  activity  after  separation 
more  Ur.  X  must  be  formed  in  the  uranium  compound  to  give 
rise  to  these  rays.  This  can  be  shown  to  be  true,  for  Ur.  X  can 


§80 
•6-60 


,10 


0      20     40      60     80     100    120    140    160 
Fio.  450.— (After  Rutherford,  Radio-activity.) 


510  RADIO-ACTIVITY 

be  separated  a  second  time  after  recovery  has  taken  place.  The 
activity  of  normal  uranium  does  not  change,  consequently, 
there  must  be  a  state  of  equilibrium  in  the  uranium  in  which  Ur. 
X  is  being  formed  at  the  same  rate  as  it  dies  away,  so  that  the 
resultant  activity  remains  constant.  This  is  borne  out  by  the 
fact  that  the  rate  of  decay  of  Ur.  X  is  the  same  as  the  rate  of 
recovery  of  the  uranium  from  which  it  was  separated.  Processes 
of  a  similar  nature  have  been  shown  to  be  continually  taking 
place  in  radium  and  actinium  compounds. 

These  facts,  along  with  a  great  deal  of  additional  evidence, 
some  of  which  we  shall  consider  later,  led  Rutherford  and  Soddy 
to  formulate  the  theory  of  successive  changes  in  radio-active  sub- 
stances. According  to  this  theory  the  different  radio-active  sub- 
stances are  gradually  undergoing  a  process  of  transformation  by 
which  they  are  changing  in  regular  succession  from  one  product 
to  another  without  the  help  of  any  outside  agency.  We  shall  see 
later  that  Th.  X,  for  instance,  is  not  lost  when  its  activity  com- 
pletely decays,  but  it  disappears  as  Th.  X  and  changes  into  another 
product  or  substance.  Most  of  these  transformation  products 
as  they  are  called  give  out  radiations  similar  to  those  we  have 
considered,  but  some  do  not  give  out  any  at  all  and  are,  conse- 
quently, called  rayless  products.  The  rates  at  which  these 
changes  take  place  vary  very  greatly  for  the  different  products, 
some  changes  only  taking  a  few  seconds  to  complete,  while 
others  extend  over  hundreds  of  years.  The  time  it  takes 
any  one  of  these  changes  to  be  half  completed  is  generally  spoken 
of  as  the  period  of  that  transformation,  as  this  time  is  usually  much 
more  easily  determined  experimentally  with  accuracy  than  the 
time  of  the  complete  change. 

Actinium  possesses  a  corresponding  active  constituent  called 
Actinium  X  with  properties  similar  to  Th.  X. 

677.  Emanations  from  Radio-active  Bodies. — The  early  experimenters 
on  thorium  observed  that  the  radiations  given  out  by  thorium  compounds 
were  very  irregular.  Rutherford  investigated  this  irregularity  and  found 
that  it  was  due  to  the  emission  of  some  sort  of  radio-active  particles  from 
the  thorium  compound.  To  these  particles  he  gave  the  name  "emanation," 
and  he  found  that  it  was  not  like  the  radiations  which  we  have  already 
considered,  but  acted  in  all  respects  like  a  gas.  It  will  diffuse  through  porous 
solids  and  through  gases  and  it  may  be  carried  away  by  a  current  of  air. 
It  ii  capable  of  ionizing  a  gas  itself  and  of  acting  on  a  photographic  plate. 


EXCITED  ACTIVITY  511 

It  does  not  itself  consist  of  ions  but  has  the  power  of  producing  ions  in  the 
gas,  for  it  may  be  passed  through  cotton  wool  or  bubbled  through  solutions 
without  losing  its  power  of  ionizing  a  gas.  This  differs  from  a  gas  ionized 
in  the  ordinary  way,  for  the  gas  will  lose  its  ions  under  these  circumstances 
while  the  emanation  does  not. 

The  emanation  is  not  affected  by  an  electric  field.  The  electric  field 
removes  the  ions  produced  by  it  but  does  not  remove  the  emanation  itself. 
The  emanation  cannot,  therefore,  consist  of  charged  particles  like  the  ions. 

Both  radium  and  actinium  compounds  give  out  an  emanation  possessing 
properties  similar  to  the  thorium  emanation,  but  as  far  as  is  known  at  present 
uranium  compounds  do  not  give  off  any  emanation. 

These  emanations  are  chemically  inactive,  not  being  affected  by  the 
strongest  reagents.  They  are  not  altered  by  being  passed  through  a  plati- 
num tube  raised  to  a  white  heat,  nor  by  being  cooled  to  the  temperature  of 
solid  carbon  dioxide.  The  emanations  can  be  condensed  when  passed 
through  a  tube  immersed  in  liquid  air.  This  is  a  very  important  and  crucial 
experiment,  proving  conclusively  the  gaseous  nature  of  the  emanation. 

Actinium  emanation  may  be  condensed  under  the  same  conditions  as 
thorium  emanations. 

If  the  emanation  be  removed  from  the  thorium,  by  drawing  off  into 
another  vessel  both  it  and  the  air  with  which  it  is  mixed,  its  activity  dies 
away  very  rapidly  with  time.  Also  if  a  quantity  of  thorium  be  placed  in  a 
closed  vessel  and  the  ionization  currrent  measured  immediately  and  at 
short  intervals  it  is  found  to  gradually  rise  and  finally  reach  a  steady  state. 
The  rate  at  which  the  current  rises  in  the  closed  vessel  is  exactly  the  same 
as  the  rate  at  which  the  separated  emanation  dies  away.  We  have  here  a 
state  of  things  similar  to  the  case  of  thorium  and  Th.  X  where  the  activity 
of  one  rises  at  the  same  rate  as  the  other  dies  away.  An  equilibrum  state 
is  reached  when  the  emanation  is  produced  as  fast  as  it  dies  away. 

The  emanation  is  not  produced  directly  by  the  thorium  but  is  a 
product  of  Thorium  X.  Rutherford  and  Soddy  have  shown  that  when  the 
Th.  X  is  separated  from  the  thorium  the  latter  does  not  give  off  any  ema- 
nation but  gradually  regains  its  emanating  power.  The  separated  Th.  X, 
however,  possesses  strong  emanating  power  but  gradually  loses  it.  These 
processes  take  place  at  exactly  the  same  rate  as  the  loss  and  regain  of  activity 
by  the  Th.  X  and  thorium  respectively,  which  we  have  already  considered. 
This  accounts  for  the  decay  of  the  Th.  X  as  it  is  continually  changing  into 
emanation.  The  emanation  and  Th.  X  are  distinct  substances  having 
distinct  properties.  These  emanations  resemble  somewhat  the  rare  gases 
found  in  the  atmosphere,  being  very  inert  chemically.  Radium  emanation 
is  now  definitely  recognized  as  an  element  having  an  atomic  weight  of  220 
and  it  has  been  given  the  name  niton. 

578.  Excited  Activity. — If  a  solid  body  be  exposed  in  a  closed  vessel  to  the 
emanations  from  radium,  thorium  or  actinium  its  surface  becomes  coated 
with  an  extremely  thin  solid  deposit  of  very  radio-active  material.  This 
active  deposit  is  invisible,  even  under  a  microscope,  but  can  be  dissolved  by 
certain  acids  and  when  the  solvent  is  evaporated  again  it  is  left  behind.  It 


512  RADIO-ACTIVITY 

emits  radiations  which  affect  a  photographic  plate  and  ionize  a  gas.  If  a 
negatively  charged  wire  be  placed  in  a  closed  vessel  containing  the  emana- 
tion the  active  deposit  is  all  concentrated  on  this  wire  instead  of  being 
distributed  on  the  interior  of  the  vessel.  By  this  means  a  very  small  wire 
may  be  made  intensely  radio-active. 

The  active  deposit  can  be  removed  from  a  wire  by  rubbing  with  sand 
paper,  but  the  quantity  deposited  is  so  extremely  small  that  no  increase  in 
weight  can  be  detected  in  a  wire  which  has  received  an  active  deposit.  This 
active  deposit  is  not  due  to  any  action  of  the  radiations  given  out  by  the 
radio-active  compound  but  is  a  direct  result  of  the  presence  of  the  emanation, 
for  when  no  emanation  is  present  no  active  deposit  is  observed  and,  in  addi- 
tion, the  amount  of  excited  activity  is  always  proportional  to  the  amount  of 
emanation  present. 

If  the  negatively  charged  wire  be  exposed  to  the  emanation  for  several 
hours  and  then  removed  and  its  activity  tested  at  intervals,  it  is  found  to 
gradually  die  away  with  time,  according  to  a  law  exactly  similar  to  that  for 
the  decay  of  the  emanations.  The  excited  activity  from  thorium  decays  to 
half  value  in  about  10.6  hours.  It  requires  time  for  the  excited  activity  to 
be  deposited  on  the  wire  and  the  deposit  increases  until  it  reaches  a  maxi- 
mum. This  rate  of  increase  is  the  same  as  the  rate  of  decrease  of  activity 
when  the  wire  is  removed  from  the  emanation.  There  must,  consequently, 
be  an  exactly  similar  process  going  on  here  as  we  observed  in  connection 
with  thorium  and  Th.  X,  and  with  Th.  X  and  the  emanation.  Just  as  Th. 
X  is  continuously  changing  into  the  emanation  the  emanation  is  gradually 
changing  into  the  active  deposit  and  this  in  turn  must  be  changing  into 
something  else. 

If  the  wire  be  exposed  to  the  thorium  emanation  for  only  a  few  minutes 
instead  of  several  hours  a  different  phenomenon  is  observed  after  removal 
of  the  wire.  Instead  of  beginning  to  decay  immediately  after  removal  the 
activity,  which  at  first  is  very  small,  gradually  increases  until  it  reaches  a 
maximum  in  about  four  hours,  and  then  it  decays  again  at  just  the  same  rate 
as  the  activity  for  a  long  exposure  decayed.  When  the  exposure  is  a  long 
one  no  initial  increase  is  observed.  Rutherford  suggested  that  the  active 
deposit,  instead  of  being  one  substance,  is  really  made  up  of  two  distinct 
substances  one  of  which  is  changing  into  the  other.  He  called  these  two 
substances  thorium  A  and  thorium  B  and  supposed  that  thorium  A  arose 
from  the  emanation  and  was  deposited  on  the  wire  and  then  changed  into 
thorium  B,  and  then  the  thorium  B  changed  into  something  else.  For  a 
short  exposure  the  deposit  will  consist  almost  entirely  of  thorium  A,  as  very 
little  has  had  time  to  change  into  thorium  B,  and  if  we  suppose  that  thorium 
A  either  gives  out  no  rays  at  all  or  rays  which  produce  a  very  small  amount 
of  ionization  compared  with  those  from  thorium  B,  then  the  activity  at  first 
will  be  very  small,  due  almost  entirely  to  the  very  small  portion  of  thorium 
B.  As  thorium  A  changes  into  thorium  B  the  activity  will  increase  until 
the  change  of  A  into  B  just  balances  the  decay  of  B.  Then,  as  more  atoms 
of  B  will  change  per  second  than  are  produced  from  A,  the  activity  will 
gradually  decay.  In  the  case  of  the  long  exposure  this  maximum  has  been 


THEORY  OF  RADIO-ACTIVE  CHANGES  513 

reached  before  the  wire  is  removed  and  tested  and,  consequently,  the  initial 
rise  is  not  observed.  Recent  investigations  show  that  this  active  deposit  is 
more  complex  than  was  at  first  supposed.  It  is  now  known  that  it  consists 
of  several  distinct  substances. 

An  examination  of  the  active  deposit  from  radium  shows  that  the  trans- 
formations taking  place  are  more  complicated  than  those  of  thorium  and 
actinium.  The  decay  curves  when  measured  by  the  a  rays  are  quite  different 
from  those  obtained  by  the  /?  or  y  rays.  The  two  latter  give  identical  curves 
showing  that  the  /?  and  f  rays  occur  together.  By  a  process  of  analysis 
similar  to  that  used  for  thorium  it  has  been  shown  that  the  active  deposit 
from  the  radium  emanation  consists  of  even  a  greater  number  of  distinct 
transformation  products  than  is  produced  by  thorium  or  actinium. 

679.  Heat   Emitted   by   Radium   and   Thorium. — Curie    and 
Laborde   discovered   that   radium   is    always   hotter   than   its 
surroundings  and  emits  heat  at  the  rate  of  100  calories  per  gram 
per  hour.     It  has  also  been  found  that  thorium  acts  similarly 
though  in  a  minor  degree.     This  is  readily  explained  by  the  high 
velocity  and  kinetic  energy  of  the  a  particles  (§572)  and  the 
readiness  with  which  they  are  absorbed  (§574).     Many  of  the 
particles  that  start  within  the  radio-active  body  are  absorbed  by 
the  body  itself  and  their  kinetic  energy  is  transformed  into  heat. 

680.  Theory  of  Radio-active  Changes. — We  have  seen  that  in 
the  radio-active  bodies  continuous  changes  from  one  substance  to 
another  are  taking  place  which  so  far  have  never  been  observed 
in  any  other  class  of  materials.    Each  of  these  substances  is 
entirely  distinct  from  the  others  and  has  distinct  physical  and 
chemical  properties.    They,  however,  gradually  decay  and  each 
one  has  a  distinct  and  definite  period  of   decay  which  dis- 
tinguishes it  from  all  the  others.     How  do  these  changes  come 
about?     The    disintegration    theory   or   theory    of   successive 
changes  furnishes  the  now  generally  accepted  explanation. 

According  to  the  theory  of  J.  J.  Thomson  atoms  may  be  con- 
sidered complex  structures  consisting  of  systems  of  positively  and 
negatively  charged  particles  in  very  rapid  rotation  and  held  to- 
gether by  their  mutual  forces  in  equilibrium.  According  to  the 
disintegration  theory  this  complex  structure  constituting  the 
atom  of  radium  (which  we  shall  take  as  a  typical  example) 
becomes  by  some  means  unstable  and  one  of  the  positively 
charged  a  particles  is  suddenly  expelled  with  great  velocity. 
The  structure  of  the  atom  which  remains  is  now  different  and 
constitutes  the  atom  of  a  new  substance,  namely,  the  emanation. 

33 


514  RADIO-ACTIVITY 

The  atoms  of  the  emanation  are  unstable  and  gradually  change 
by  the  expulsion  of  another  a  particle,  leaving  a  new  structure, 
namely,  the  atom  of  radium  A,  and  the  process  is  continued 
throughout  the  successive  changes.  The  processes  are  not  iden- 
tical in  all  instances,  for  in  some  cases  an  a  particle  alone  is  ex- 
pelled, but  in  others  /?  particles  are  expelled  accompanied  by  f 
rays,  while  in  others  all  three  types  are  given  out. 

Why  do  these  atoms  suddenly  become  unstable  and  break  up 
without  any  apparent  cause?     Several  explanations  have  been 
offered  to  account  for  this,  but  the  most  probable  one  seems  to  be 
that  if  this  system  of  charged  particles,  of  which  the  atom  prob- 
ably consists,  is  in  rapid  rotation  it  must  be  radiating  energy,  and 
when  sufficient  energy  has  been  radiated  the  mutual  forces  of  the 
system  no  longer  balance  and  one  or  more  of  the  particles  escape  and 
cause  disintegration.     These  atoms  have  an  independent  existence 
and  distinct  physical  and  chemical  properties,  but  they  differ  from 
the  atoms  of  ordinary  non-radio-active  elements  in  the  fact  that 
they  are  not  permanent.     To  distinguish  them  from  ordinary 
atoms  the  term  metabolon  has  been  suggested  as  a  suitable  name. 
A  few  of  these  transformation  products  do  not  emit  any  rays  at 
all  and  the  change  from  them  into  the  succeeding  substance 
apparently  takes  place  without  the  expulsion  of  any  particles. 
These  so-called  rayless  changes  may  be  explained  in  either  of  two 
ways.     The  new  product  may  be  formed  in  this  case  simply  by  a 
rearrangement  of  the  system  of  charged  particles,  but  not  with 
sufficient  violence  to  expel  any  of  the  system,  or  it  may  be  pro- 
duced by  the  expulsion  of  one  or  more  particles,  but  with  a 
velocity  too  slow  to  ionize  the  gas.     It  has  been  shown  that  when 
the  velocity  of  the  a  particle  falls  below  109  cms.  per  second  it 
ceases  to  ionize  the  gas,  and  consequently  an  a  particle  expelled 
with  a  velocity  below  this  minimum  would  escape  detection 
since  no  ions  would  be  produced. 

This  latter  hypothesis  suggests  that  all  matter  may  possibly 
be  undergoing  a  slow  change  in  a  similar  manner,  and  that  the 
reason  this  change  has  been  observed  only  in  the  so-called  radio- 
active bodies  and  not  in  other  non-radio-active  bodies  is  that  in 
the  case  of  the  radio-active  bodies  the  charged  particles  are  expelled 
with  sufficient  violence  to  ionize  the  gas  while  in  other  bodies  they 
may  be  expelled  but  not  With  sufficient  velocity  to  produce  ions . 


RADIO-ACTIVE  ELEMENTS 


515 


581.  Radio-active  Elements. — The  following  table  contains  a 
summary  of  all  the  active  products  at  present  known.  On 
account  of  the  incomplete  state  of  the  subject  this  list  will  in  all 
probability  undergo  in  the  future  slight  changes  as  a  result  of 
further  investigation. 

TABLE  OF  RADIO-ACTIVE  ELEMENTS 


Radio-active  products 

Transfor- 
mation 
period 

Nature 
of  rays 
emitted 

Radio-active 
products 

Transfor- 
mation 
period 

Nature 
of  roys 
emitted 

5X10» 

1  8X1019 

1  ^Uranium  Y 

years 

ft 

years 

No 

Uranium  Xi  
Uranium  Xj 

24.6  days 
1  15  min. 

P 
0 

Mesothorium  II  

6.2  hours 
2.02  years 

rays 
a 

1 

Uranium  2 

2X10S 

a 

Thorium  X  

3.64  days 

a 

4 

Ionium             .                   .  . 

2X105 

a 

Thorium  emanation 

54  S63. 

a 

i 
Radium   

years' 
1730  years 

a,  0 

Thorium  A  

.14  sec. 

a 

3.85  days 

Thorium  B     

10.6  hours 

0 

3  min 

Thorium  Ci  

60  min. 

a,  /? 

Radium  B  

28.7  min. 

p 

*ThoriumD 

3.1  min. 

/? 

Radium  Ci 

19  5  min. 

a,  ft 

Thorium  Cj  

10-"  sec. 

a 

1 

Is*  Radium  Cz 

1  4  min. 

0 

No 

Radium  C'. 

10~s  sec. 

a 

Radio-actinium  .... 

19.5  days 

rays 
a 

Radium  D  

16.5  years 

$ 

Actinium  X 

11.  4  days 

a,  13 

Radium  E 

5  days 

ft 

Actinium  emanation 

3.9  see. 

a 

Radium  F  (Polonium)  

136  hours 

a 

Actinium  A  
Actinium  B  

.002  sec. 
36.1  min. 

a 

ft 

Actinium  C.  .  . 

2.  15  min. 

a 

1 
Actinium  D  

4.71  min. 

0,  7 

feVery  recently  another  product  called  Ur  Y  has  been  discovered.  It  has  a  period  of  1.5 
days  and  emits  soft  /3  rays  and  probably  a  rays.  It  is  considered  to  be  a  lateral  disintegra- 
tion product  of  uranium. 

References. 

THOMSON'S  Conduction  of  Electricity  through  Gases. 

STRUTT'S  The  Becquerel  Rays  and  Other  Properties  of  Radium. 

RUTHERFORD'S  Radio-activity. 

RUTHERFORD'S  Radio-active  Transformations. 

McCLUNo's  Conduction  of  Electricity  through  Oases  and  Radio-activity. 


SOUND 

BY  A.  WILMEB  DUFF,  D.  So. 

Professor  of  Physics  in  the  Worcester  Polytechnic  Institute, 
Worcester,  Mass. 

NATURE  AND  PROPAGATION  OF  SOUND 

682.  Sources  of  Sound. — On  hearing  a  sound  we  instinctively 
think  of  its  origin  and  we  are  usually  able  to  trace  it  to  some  body 
which  we  call  the  source  of  the  sound.  To  ascertain  how  a  body 
produces  a  sound  we  may  take  a  body  that  can  be  kept  sounding 
while  it  is  being  observed.  If  a  large  bell  be  made  to  produce 
sound  by  striking  it,  or  a  glass  jar  by  stroking  it  with  a  violin 
bow,  a  light  pendulum  hung 
against  the  bell  or  jar  will  be 
kicked  away  at  each  contact.  A 
metal  rod  clamped  at  the  middle 
and  sounded  by  rubbing  one- 
half  of  it  with  a  rosined  cloth  or  ^^f^  \\/  / 
glove  will  give  violent  blows  to 
a  pendulum  hung  in  contact  with 
the  other  end.  If  a  violin  be  FIQ  451 
held  in  the  hand  and  one  of  the 

strings  be  plucked  or  stroked  by  a  bow,  the  hand  will  tell  us 
that  the  wooden  body  of  the  violin  is  vibrating,  and,  while  the 
vibrations  of  the  body  cannot  be  seen,  those  of  the  string  are 
clearly  visible. 

In  all  cases  in  which  the  action  can  be  observed  closely  enough, 
sounding  bodies  are  found  to  be  in  a  state  of  vibration.  In  many 
cases  the  vibrations  are  so  small  or  cease  so  quickly  that  they 
cannot  be  detected.  But  there  is  no  doubt  that  the  head  of  a 
hammer  vibrates  for  a  moment  when  it  strikes  a  nail  and  the 
board  into  which  the  nail  is  being  driven  also  vibrates  for  a 
moment. 

683.  Media  in  which  Sound  Travels. — It  is  well  known  that  the 
clearness  with  which  distant  sounds  are  heard  depends  on  the 

517 


518  SOUND 

state  of  the  atmosphere  and  the  direction  of  the  wind.  Hence  the 
air  is  the  ordinary  medium  of  transmission.  Sound  will  not 
travel  in  a  vacuum,  as  can  be  readily  shown  by  placing  an  electric 
bell  under  the  receiver  of  an  air  pump.  The  sound  will  diminish 
as  the  air  is  removed,  but  it  will  be  restored  if  the  air  or  any  other 
gas  is  allowed  to  enter. 

Sound  is  also  transmitted  by  liquids  and  solids.  If  two  stones 
be  struck  together  under  water  a  loud  sound  will  be  heard  by  an 
ear  held  beneath  the  surface.  A  watch  placed  on  one  end  of  a 
long  table  can  be  heard  by  an  ear  pressed  against  the  other  end 
of  the  table.  Miners  imprisoned  by  an  accident  in  a  mine  some- 
times send  signals  to  the  outer  world  by  blows  of  their  picks  on 
the  rock.  The  approach  of  a  distant  train  or  the  galloping 
of  horses  can  be  heard  by  an  ear  pressed  against  the  ground. 
Beethoven,  who  was  deaf,  heard  some  of  his  own  compositions 
only  by  means  of  a  stick,  one  end  of  which  rested  against  the 
sounding  board  of  the  piano  while  the  other  was  pressed  against 
his  teeth. 

584.  The  Nature  of  Sound. — Since  the  sources  of  sound  are 
vibrating  bodies,   it  follows  that  sound  travelling  through   a 
^_^^  ^_^^  ^_^        medium  must  be  a  wave 

f        >,          j       \,          f       \       motion.     That  it  does  not 
V_ s  N — x  consist,  like  odors,  of  par- 

ticles transmitted  from  the 

source  °f  sound  is  shown 

Illlllilllllllllilllllllllllllll       by  the  fact  that  sounds  of 

great  intensity   can   come 

FIG.  452. 

to  us  through  a  layer  of 
smoky  air  without  bringing  the  smoke  with  it. 

Wave  motions  may  be  either  transverse,  like  the  to  and  fro 
motions  of  a  cord,  or  longitudinal,  like  the  compressions  and 
extensions  of  a  spiral  spring,  or  they  may  be  a  mixture  of  both, 
as  in  the  case  of  water  waves.  Now  a  gas  offers  elastic  resistance 
to  compression  and  can,  therefore,  transmit  longitudinal  waves; 
but  it  offers  no  sustained  resistance  to  changes  of  shape,  or  shears, 
and  it  cannot,  therefore,  transmit  transverse  elastic  waves. 
Hence  sound  waves  are  longitudinal  waves  or  waves  of  compres- 
sion and  dilatation. 

To  represent  sound  waves  graphically  we  make  use  of  the 


NATURE  AND  PROPAGATION  OF  SOUND  519 

device  explained  in  §238,  that  is,  we  take  the  axis  of  abscissae 
in  the  direction  in  which  the  waves  are  travelling  and  draw  ordi* 
nates  to  represent  the  displacements  of  the  air  particles  at  the 
corresponding  points,  although  the  displacements  are  really  in 
the  direction  of  propagation.  These  actual  displacements 
are  in  reality  excessively  minute,  ranging  from  2  or  3  mm. 
in  the  loudest  sounds  to  about  10~7  mm.  in  sounds  that  are  just 
audible.  Hence  the  ordinates  are  drawn  on  a  greatly  enlarged 
scale.  The  forward  motion  of  the  wave  is  represented  to  the 
mind  by  supposing  the  curve  to  move  forward  with  the  velocity 
of  the  wave.  The  curve  crosses  the  axis  of  abscissae  at  the  middle 
of  a  condensation  or  of  a  rarefaction  and  the  maximum  ordinate 
occurs  at  a  place  of  no  condensation  or  rarefaction.  Sound 
waves  are  sometimes  represented  by  similar  curves  the  ordinates 
of  which  represent  the  degrees  of  condensation  (positive  or 
negative)  at  points  in  the  wave.  Such  curves  are  half  a  wave 
length  ahead  of  (or  behind)  those  that  represent  displacements. 

That  sound  has  all  the  properties  characteristic  of  wave  motions 
in  general  will  appear  in  the  paragraphs  which  immediately  follow. 

685.  Velocity  of  Sound. — When  a  man  is  seen  chopping  a  log 
or  hammering  a  nail  at  a  great  distance,  the  sound  of  each  blow 
is  not  heard  until  some  time  after  the  blow  is  seen.  Steam  may 
be  seen  to  issue  from  a  distant  whistle  before  the  sound  is  heard. 
Fire-alarm  whistles  sounded  simultaneously  may  be  heard 
separately  if  the  observer  is  not  at  equal  distances  from  the 
stations.  Lightning  precedes  thunder.  From  these  facts  it  is 
evident  that  sound  travels  at  a  definite  rate  which  can  be  meas- 
ured. That  this  rate  is  not  as  great  as  the  velocity  of  shells 
fired  by  modern  high  power  guns  is  shown  by  the  experiences  of 
soldiers  in  the  trenches  in  the  European  war.  The  sound  made 
by  a  passing  shell  was  heard  before  that  of  the  firing  of  the  gun 
arrived. 

The  velocity  of  sound  has  been  determined  by  methods  sug- 
gested by  some  of  the  above  experiences.  The  most  accurate  of 
such  direct  determinations  were  made  near  Utrecht  in  1823 
and  near  Paris  in  1822,  and  gave  a  mean  value  of  about  341  meters 
per  second  at  16°G.  The  sounds  observed  were  those  of  the 
firing  of  cannon  at  great  distances  and  care  was  taken  to  reduce 
the  effect  of  wind  by  alternate  observations  of  sounds  travelling 


520  SOUND 

in  opposite  directions.  The  velocity  of  sound  in  water  was  found 
in  1827  by  means  of  bells  sounded  beneath  the  water  of  Lake 
Geneva  in  Switzerland.  But  we  shall  see  later  that  there  are 
more  readily  available  laboratory  methods  for  finding  the  velocity 
of  sound  in  gases,  liquids,  or  solids. 

That  sounds  of  different  pitch  travel  at  practically  the  same 
rate  is  shown  by  the  fact  that  music  made  by  a  band  can  be  heard 
as  music  at  a  considerable  distance;  for  if  notes  sounded  at  the 
same  time  were  not  heard  simultaneously  both  harmony  and 
melody  would  be  distorted.  There  is  some  evidence  that  very 
loud  sounds  may  travel  at  somewhat  greater  velocity  than  others, 
but  the  difference,  if  it  exists,  is  slight. 

686.  Formula  for  the  Velocity  of  Sound. — From  the  principles 
of  Mechanics  applied  to  waves  of  compressions  and  rarefactions 
Newton  derived  the  formula  which  has  been  stated  in  §251,  namely, 

IE 


p  being  the  density  of  the  medium  and  E  its  elasticity.  Now  the 
elasticity  of  a  gas  at  constant  temperature,  or  its  isothermal 
elasticity,  is  equal  to  the  pressure,  p  ( §223) .  But  if  we  substitute 
pt  expressed  in  absolute  units  (dynes  per  square  centimeter) 
in  the  above  we  get  a  result  which  is  much  too  small,  as  Newton 
found.  This  difficulty  was  not  removed  until  Laplace  pointed 
out  that  the  temperature  must  be  momentarily  elevated  in  a 
compression  and  depressed  in  a  rarefaction  as  a  train  of  sound 
waves  passes,  and  that  E  should  therefore  betaken  as  the  adiabatic 
elasticity,  which,  as  we  have  seen  (§346)  is  up,  where  *  is  the  ratio 
of  the  specific  heats  of  the  gas  at  constant  pressure  and  at 
constant  volume  respectively.  With  this  condition  the  formula 

becomes 

, — 


and  this  is  found  to  agree  with  experimental  results. 

From  this  formula  it  is  evident  that  the  velocity  of  sound  in  a 
gas  is  independent  of  the  pressure  or  density,  provided  the  tem- 
perature is  constant,  since,  in  accordance  with  Boyle's  law,  p 
is  proportional  to  p.  Thus  at  the  same  temperature  the  velocity 
is  the  same  at  the  top  of  a  mountain  as  in  a  valley.  In  gases  of 


NATURE  AND  PROPAGATION  OF  SOUND  521 

different  densities  the  velocities  are  inversely  as  the  square  roots 
of  the  densities.  Thus  the  velocity  in  hydrogen  is  four  times  that 
in  oxygen.  Since  the  density  of  water  vapor  is  less  than  that  of 
air,  the  presence  of  water  vapor  in  air,  at  given  atmospheric 
pressure,  causes  a  slight  decrease  in  the  -velocity  of  sound. 

From  the  above  formula  we  can  also  find  how  the  velocity  of 
sound  depends  on  the  temperature  of  a  gas.  For  from  the 
general  gas  law  (§280),  since  density  varies  inversely  as  volume, 
we  see  that  p/p=  (pjp0) (1  -I- at) .  Hence 


Po 

I  •»" 
where  v0=     ^  is  the  velocity  at  0°C. 

\     Po 

VELOCITY  OF  SOUND  IN  DIFFERENT  SUBSTANCES  (IN  METERS  PER 
SECOND)  AT  0°C. 

Air 332    Steel 4975 

Hydrogen 1268    Lead 1420 

Carbon  dioxide! 261     Glass 4860 

Freshwater 1435    Pine  wood 3300 

Sea  water 1454     Walnut  wood 4800 

Mercury 1484    India  rubber 5000 

687.  Mechanical  Effects  of  Sound  Waves. — Sound 
waves  produce  mechanical  effects  by  which  they 
can  be  detected  and  to  some  extent  measured.  A 
very  simple  and  useful  detector  is  a  "sensitive 
flame,"  that  is,  a  long  slender  gas  jet  issuing  from 
a  very  fine  nozzle  (a  glass  tube  drawn  out  to  a 
fine  point)  under  steady  pressure.  If  the  flow  is 
regulated  until  the  flame  is  just  about  to  flare  or 
become  unsteady,  sounds  of  high  pitch  falling  on 
the  orifice  will  cause  instability  and  the  flame  will 
shorten  greatly  and  "roar"  like  an  ordinary  gas 
flame  when  the  stopcock  is  too  wide  open.  A  hiss 
or  the  rattling  of  a  bunch  of  keys  is  especially  ef- 
fective. A  Welsbach  flame  turned  very  low  can  be 
made  very  sensitive  to  high  pitched  sounds  (like 
those  on  the  highest  string  of  a  violin). 

A  so-called  "manometric  flame"  is  a  gas  jet  fed 
by  gas  which  passes  through  a  small  box  one  side 


522 


SOUND 


of  which  is  of  thin  rubber.  Sound  waves  falling  on  the  rubber 
produce  variations  of  pressure  in  the  gas  and  corresponding  vari- 
ations in  the  height  of  the  flame.  If  the  latter  be  viewed  in  a  ro- 
tating mirror,  a  band  of  curved  outline  corresponding  to  the  pe- 
culiarities of  the  sound  will  be  seen. 


Fia.  455 


Fia.  454. 

The  vibrations  of  a  disk  on  which  sound  waves  fall  can  be 
made  visible  by  reflecting  a  beam  of  light  from  a  small 
mirror  connected  with  the  disk.  This  method  has  been  very 
successfully  used  by  Professor  D.  C.  Miller  in  his  "phonodeik," 

a  fine  fiber  /  attached  at  one 
end  to  a  thin  glass  plate  p  is 
-)  wrapped  around  a  tiny  spindle 
which  carries  the  mirror  m. 
A  record  obtained  by  Miller 
will  be  given  later  (Fig.  465)  • 

The  effect  of  sound  waves  on  the  ear  consists  of  a  vibration  of  the  ear 
drum  produced  by  the  alternating  pressure  of  the  waves.  Their  effect  on 
the  disk  of  a  telephone  is  similar.  The  vibrations  of  the  disk  produce 
alternating  electric  currents  in  the  line  ( §525)  and  these  can  be  shown  by  a 
vibration  galvanometer  (§438)  or  oscillograph  (§524),  or  they  can  be  rectified 
by  a  crystal  rectifier  ( §545)  and  shown  by  an  ordinary  galvanometer. 

Sustained  waves  of  any  kind  carry  a  steady  stream  of  energy  from  the 
source  and  falling  on  a  surface  produce  a  sustained  pressure.  In  the  case  of 
sound  waves  this  pressure  is  small.  It  can  be  shown  by  concentrating  the 
loud  sound  produced  by  a  stream  of  sparks  from  an  induction  coil  by  means 
of  a  large  metallic  mirror.  A  small  vane  similar  to  that  of  a  radiometer  placed 
at  the  focus  will  be  set  in  rotation.  Waves  falling  on  a  small  disk  in  a  reso- 
nating tube  (Fig.  456)  tend  to  set  it  at  right  angles  to  the  stream  by  an  action 
somewhat  similar  to  that  described  in  §204.  This  action  has  been  used  in 


NATURE  AND  PROPAGATION  OF  SOUND 


523 


certain  investigations  on  the  energy  flow  in  sound  waves,  by  Lord  Rayleigh 
and  others.  No  instrument  of  general  usefulness  in  the  measurement  of 
sound  intensity,  that  is,  of  the  energy  flow,  has  yet  been  devised. 

588.  Photographs  of  Sound  Waves. — Sound  waves  can  be 
photographed  by  a  method  very  analogous  to  the  method  used 


Fia.  456. 


in  projecting  ripples  (§256).  The  waves  pass  in  front  of  a  photo- 
graphic plate  in  the  dark;  a  single  spark  from  a  distant  Ley  den 
jar  illuminates  the  plate  for  a  moment  and  leaves  a  record  of 
the  condensations  and  rarefactions  owing  to  the  refraction  of  the 
light  as  it  passes  through  the  sound  wave.  Fig.  457  is  such  a 


Fia.  457. 

record  (by  Foley  and  Souder)  showing  the  reflection  of  a  spher- 
ical wave  of  sound  by  a  plane  surface.     (See  also  §691). 

589.  Reflection  of  Sound. — Sound,  like  other  wave  motions,  is 
reflected  when  it  falls  on  a  suitable  surface,  and  in  the  process 
it  follows  the  same  laws  as  light  and  radiant  heat,  namely,  the 
reflected  and  incident  rays  lie  on  opposite  sides  of  the  normal  to 


524  SOUND 

the  surface  and  make  equal  angles  with  it.  The  identity  of  these 
laws  for  sound  and  light  can  be  shown  by  means  of  two  large 
concave  metallic  mirrors  placed  opposite  each  other  with  their 
axes  in  the  same  line.  In  front  of  each  mirror  there  is  a  point, 


Fia.  458. 

its  principal  focus,  such  that  a  bright  light  placed  at  one  focus 
produces  a  bright  image  at  the  other.  When  these  points  have 
been  found,  a  source  of  sound  of  high  pitch  placed  at  one  focus 
will  produce  an  intense  sound  at  the  other  focus,  as  may  be  seen 
by  its  action  on  a  sensitive  flame.  If  a  singing  arc  light  be 
placed  at  one  focus,  the  light,  heat,  and  sound  will  be  focussed 
simultaneously  by  the  other  mirror,  as  may  be  ascertained  by 
means  of  a  small  screen,  a  thermopile  (§333),  and  a  sensitive 
flame  respectively. 

Reflection  of  sound  gives  rise  to  echoes.  The  echo  from  a 
large  reflecting  surface,  such  as  the  side  of  an  isolated  house,  is 
heard  most  distinctly  when  the  observer  is  in  a  line  perpendicular 
to  the  surface,  and  it  decreases  rapidly  as  he  moves  away  from 
that  line.  Most  of  the  sound  heard  in  an  auditorium  is  reflected 
sound.  This  will  be  referred  to  more  fully  later. 

690.  Refraction  of  Sound. — Waves  are  refracted,  or  their  line 
of  propagation  is  bent  when  they  pass  obliquely  from  one  medium 
into  another  in  which  the  velocity  is  different  (§255).  The 
refraction  of  sound  cannot  be  shown  satisfactorily  on  a  small 
scale  by  lecture  or  laboratory  apparatus,  but  it  takes  place  on  a 
large  scale  in  nature.  The  chief  causes  of  such  refraction  are 
winds  and  variations  of  temperature  in  the  atmosphere. 

The  velocity  of  a  wind  is  less  nearer  the  surface  of  the  earth 
than  higher  up,  since  near  the  surface  it  is  retarded  by  the  fric- 
tional  resistance  of  the  surface.  When  sound  is  travelling  in  the 
same  direction  as  the  wind  its  resultant  velocity  is  greater  above 


NATURE  AND  PROPAGATION  OF  SOUND 


525 


than  below.  Hence  the  waves,  which  always  travel  at  right 
angles  to  their  fronts,  are  tilted  forward,  or  the  direction  of 
their  motion  is  deflected  downward  (Fig.  459,  B) .  The  opposite 
effect  takes  place  when  the  sound  is  travelling  against  the 
wind  (Fig.  459,  A).  This  explains  why  sound  is  better  heard 
with  the  wind  than  against  it  and  why  it  is  an  advantage  in  the 
latter  case  to  listen  from  an  elevation. 

Similar  effects  result  from  the  temperature  being  different  at 
different  heights^  in  the  atmosphere.  Usually  on  fine  days  the 
temperature  is  lower  at  an  elevation  and  the  velocity  of  sound  is 
less  there.  Thus  the  waves  are  deflected  upward  as  in  Fig.  459,  A 
and  hearing  near  the  surface  is  poor.  At  night  or  near  sunrise 
or  sunset  the  surface  is  colder  and  the  gradient  of  temperature  is 
less  and  sounds  are  heard  better.  Moreover  on  a  hot  day  the 
air  is  heated  irregularly  by  contact  with  parts  of  the  surface  at 


FIQ.  459. 

different  temperatures  and  there  are  numerous  irregular  con- 
vection currents  in  the  atmosphere.  These  cause  irregular 
reflections  and  refractions  of  sound  as  in  the  similar  case  of  light 
passing  over  a  heated  stove.  These  effects  also  help  us  to  under- 
stand why  distant  sounds  are  frequently  heard  most  clearly 
before  a  storm. 

691.  Diffraction  of  Sound. — Diffraction  means  the  bending  of 
waves  around  obstacles  (§256).  It  prevents  the  formation  of 
sharp  shadows.  The  amount  of  diffraction  in  any  case  depends 
on  the  linear  dimensions  of  the  obstacle  compared  with  the  wave 
length  of  the  sound.  A  hill  casts  a  fairly  definite  sound  shadow 
because  it  is  large  in  comparison  with  the  wave  length.  Cases 
have  been  known  in  which  houses  in  the  shadow  of  a  hill  have 
suffered  no  damage  from  very  loud  sounds,  such  as  the  explosion 
of  a  powder  magazine  or  the  firing  of  cannon,  while  the  windows 
of  equally  distant  houses  not  in  the  shadow  were  broken  by  the 


526  SOUND 

impact  of  the  sound  waves.  But  small  obstacles,  such  as  trees 
and  posts,  cast  no  observable  sound  shadows,  except  when  tested 
by  wave  of  very  short  length.  The  human  head  is  of  sufficient 
size  to  cast  something  of  a  shadow  for  sounds  of  high  pitch.  If 
the  sound  is  to  the  left  or  the  right  of  the  observer,  one  ear  is  in 
this  partial  shadow  and  from  the  relative  intensities  as  heard  by 
the  two  ears  we  judge  of  the  direction  of  the  source.  Long  waves 
produce  about  equal  intensities  at  the  two  ears,  but  at  one  ear 
the  phase  of  the  waves  is  later  than  at  the  other  and  it  has  been 
shown  that  it  is  by  this  slight  difference  that  the  mind  uncon- 
sciously judges  of  the  direction  of  the  source  of  the  sound. 

692.  The  Phonograph  and  the  Grammophone. — When  we  speak 
against  the  middle  of  a  thin  elastic  disk  such  as  is  used  in  the 
telephone  transmitter,  the  disk  vibrates  backward  and  forward  in 
unison  with  the  sound  waves.  In  the  phonograph  these  vibra- 
tions are  recorded  on  a  drum  of  hard  wax  that  is  kept  in  rotation 
behind  the  vibrating  disk.  A  short  needle  attached  to  the  vibrat- 
ing disk  presses  on  the  moving  drum  and  the  vibrations  cause 
it  to  plough  a  fine  furrow  in  the  wax.  Thus  the  furrow  is  a  record 
of  the  vibrations  of  the  disk.  The  sound  can  be  reproduced  by 
allowing  the  needle  to  travel  again  along  the  furrow.  It  is  thus 
pushed  up  and  down,  following  very  closely  the  motion  that 
produced  the  furrow,  so  that  it  causes  the  disk  to  repeat  its 
original  vibrations  and  the  disk  therefore  reproduces  the  original 
sound.  (It  is  found  better  to  use  different  disks  and  needles  for 
recording  and  reproducing.) 

The  principle  of  the  grammophone  is  essentially  the  same,  but 
the  needle  is  connected  to  the  center  of  the  vibrating  disk  by  a 
lever  in  such  a  way  that  it  moves  sidewise  and  not  up  and  down. 
It  thus  produces  a  transverse  furrow  on  a  rotating  disk. 

MUSICAL  SOUNDS 

593.  Characteristics  of  Musical  Sounds. — The  ear  is  remarkably 
acute  in  distinguishing  minute  differences  in  sounds.  In  doing 
so  it  takes  note  of  three  fundamental  properties  in  which  sounds 
differ,  namely,  loudness,  pitch,  and  quality.  These  we  may  call 
the  three  characteristics  of  musical  sounds.  They  and  the 
direction  from  which  the  sound  comes  are  all  that  the  ear  can 


MUSICAL  SOUNDS  527 

tell  us  about  it.  From  these  the  mind  can,  as  a  result  of  long 
experience,  draw  very  rapid  conclusions  as  to  the  nature  of  the 
source.  The  words  loudness,  pitch,  and  quality  need  no  defini- 
tion, as  we  are  more  familiar  with  them  than  with  the  terms  which 
might  be  used  in  defining  them.  But  we  must  remember  taht 
the  words  stand  for  sensations.  Sound  waves  must  have  cor- 
responding characteristics  that  account  for  these  differences  in 
the  sensations  of  sound. 

694.  Loudness. — If  we  strike  a  bell,  a  drum,  or  a  violin  string 
very  gently,  the  vibrations  of  the  instrument  will  be  of  small 
amplitude  and  the  sound  will  be  weak,  but  a  stronger  blow  will 
produce  vibrations  of  greater  amplitude  and  the  sound  will  be 
louder.  Now  it  is  evident  that  the  amplitude  of  the  air  vibrations 
is  greater  the  greater  the  vibrations  of  the  source.  Hence  the 
loudness  of  the  sound  heard  depends  on  the  amplitude  of  the 
vibrations  in  the  waves.  The  same  conclusion  is  reached  by 
considering  that  the  sound  heard  is  weaker  the  farther  the  hearer 
is  from  the  source.  The  energy  that  falls  on  the  ear  must  be  less 
at  greater  distances  and  therefore  the  amplitude  of  vibration 
must  be  less. 

The  intensity  of  sound  waves  means,  as  in  the  case  of  waves  of 
any  kind  (§259)  the  flow  of  energy  per  unit  time  per  unit  area 
perpendicular  to  the  direction  in  which  the  waves  are  travelling. 
For  simple  harmonic  waves  of  a  given  frequency  the  intensity  is 
proportional  to  the  square  of  the  amplitude  (§§112,  259);  it  is 
also  proportional  to  the  square  of  the  frequency.  Hence  in 
the  case  of  strictly  spherical  waves  spreading  from  a  point  source 
the  intensity  would  vary  inversely  as  the  square  of  the  distance 
provided  the  waves  were  not  damped  (§259).  But,  owing  to 
the  presence  of  reflecting  surfaces,  sound  waves  produced  under 
ordinary  circumstances  do  not  remain  spherical.  There  is  also 
some  damping,  due  to  viscosity  of  the  medium,  which  may  have 
considerable  effect  for  great  distances  of  transmission  out  of  doors, 
and,  as  already  noted,  lack  of  homogeneity  of  the  atmosphere 
causes  dissipation.  For  these  reasons  the  inverse  square  law 
is  not  applicable  to  sound  waves  under  ordinary  conditions  of 
hearing. 

The  intensity  of  sound  waves  and  the  loudness  of  the  sensation 
they  produce  are  related,  but  the  relation  is  not  a  simple  one. 


528  SOUND 

The  loudness  of  the  sensation  also  depends  on  the  pitch  of  the 
sound.  But  such  questions  belong  to  Psychology  rather  than 
Physics. 

695.  Pitch. — The  sound  of  a  toy-whistle  is  of  high  pitch  or 
shrill,  that  of  an  automobile  horn  is  of  medium  pitch,  while  the 
tones  of  a  church  bell  are  deep  or  of  low  pitch.  The  physical 
cause  of  these  differences  of  pitch  may  be  shown  by  comparing 
the  different  sounds  that  can  be  produced  by  drawing  a  card  along 
the^teeth  of  a  comb.  If  it  be  drawn  slowly  a  sound  of  low  pitch 
will  be  heard,  but  if  it  be  drawn  as  rapidly  as  possible  the  sound 
will  be  of  high  pitch.  Every  time  the  card  slips  off  one  tooth  and 
strikes  the  next  an  impulse  is  given  to  the  air.  The  more  numer- 
ous these  impulses  per  second,  that  is,  the  more  air-waves  are 
started  per  second,  the  higher  the  pitch  of  the  sound  will  be. 
Similar  results  are  produced  when  a  machine  saw  cuts  a  board. 
The  rise  and  fall  of  the  pitch  of  the  sound  is  due  to  the  variations 


Fio.  460.— Savart'a  wheel.  Fia  .  461. — Tuning-fork  record. 

of  the  speed  of  the  saw.  Savart's  toothed  wheel  for  showing 
the  cause  of  the  pitch  of  sounds  illustrates  the  same  principle. 

If  a  phonograph  be  driven  at  lower  than  its  normal  speed  the 
pitch  of  every  note  will  be  lowered  in  correspondence  with  the 
decrease  of  frequency  of  the  impulses  imparted  to  the  membrane. 

A  tuning-fork  with  a  small  spring  stylus  attached  to  a  prong 
can  be  made  to  inscribe  its  vibrations  on  a  revolving  lamp- 
blacked  drum  or  on  a  lamp-blacked  sheet  of  glass  drawn  beneath 
the  fork.  It  is  found  that  for  a  certain  speed  of  drum  or  glass 
the  number  of  waves  recorded  is  greater  the  higher  the  pitch  of 
the  fork.  When  this  experiment  is  made  with  sufficient  care 
the  frequency  of  the  fork  can  be  accurately  measured. 

From  the  above  we  conclude  that  the  pitch  of  a  sound 
depends  on  the  frequency  of  the  vibrations  in  the  sound  wave. 


MUSICAL  SOUNDS 


529 


Now  we  have  already  seen  that  there  is  a  simple  relation  between 
frequency  and  wave  length  in  a  medium;  they  vary  in  inverse 
ratio,  that  is,  the  longer  the  waves  the  less  the  frequency.  Hence 
we  may  also  say  that  the  pitch  of  a  sound  depends  on  the  length  of 
the  sound  wave. 

It  must  not  be  concluded  from  the  above  that  a  regular  suc- 
cession of  air-waves  produces  the  sen- 
sation of  sound,  no  matter  what  the 
frequency.  The  enormously  rapid  vi- 
brations of  the  wings  of  a  mosquito 
produce  a  sound  of  very  high  pitch 
and  the  slower  vibrations  of  a  hum- 
ming-bird's wings  produce  a  sound  of 
low  pitch;  but  no  note  of  definite  pitch 
is  produced  by  the  flight  of  a  swallow.  In  fact,  to  produce 
sounds  of  definite  pitch  the  frequency  of  the  waves  must  not 
be  less  than  about  20  per  second.  On  the  other  hand  when 
their  frequency  exceeds  about  20,000  per  second  air-waves  do  not 
produce  the  sensation  of  sound  at  all,  though  their  existence  may 


Fio.  462. — Disk  syren. 


Fia.  463. — Recording  syren. 

be  shown  by  their  action  on  a  sensitive  flame  and  other  means. 
The  upper  limit  of  frequency  is  different  for  different  persons 
and  is  lowered  by  advancing  age  and  by  disease  of  the  ear. 

The  syren  is  an  instrument  for  producing  sounds  of  definite 
pitch  by  a  succession  of  puffs  of  air  following  one  another  in  rapid 

34 


530  SOUND 

succession.  In  the  simplest  form  of  syren  a  circular  disk  with 
circular  rings  of  holes  is  driven  at  a  high  speed  and  a  puff  is  pro- 
duced every  time  a  hole  comes  opposite  the  end  of  a  tube  through 
which  air  is  driven  under  pressure.  When  the  frequency  of 
succession  of  the  puffs  is  great  enough  to  produce  a  note,  the  pitch 
can  be  raised  by  increasing  the  speed  of  the  disk  or  by  transferring 
the  tube  to  a  ring  in  which  there  is  a  larger  number  of  holes. 
In  the  more  complete  form  of  syren  the  rotation  is  produced  by 
the  compressed  air  as  it  escapes  from  a  box  the  cover  of  which  is 
a  disk  with  a  ring  of  holes  corresponding  to  those  in  the  rotating 
disk.  The  holes  in  the  rotating  disk  and  those  in  the  fixed  disk 
slope  in  opposite  directions  and  the  jets  of  air  impinge  obliquely 
on  the  sides  of  the  holes  of  the  rotating  disk  thus  causing  it  to 
rotate.  The  frequency  of  the  note  is  found  by  means  of  a  suitable 
speed  counter  geared  to  the  rotating  disk. 

696.  Doppler's  Principle. — When  an  observer  is  in  motion  to- 
ward a  source  of  sound,  the  pitch  of  the  note  heard  is  higher 


FIQ.  464. 

than  when  he  is  at  rest.  If  the  hearer  is  in  motion  away  from 
the  source,  he  hears  a  lower  note  than  when  he  is  at  rest.  Similar 
results  follow  when  the  source  is  in  motion  toward  or  away  from 
the  observer.  The  pitch  of  the  gong  of  a  fire  engine  or  of  the 
whistle  of  a  locomotive  is  higher  when  the  source  is  approaching 
the  hearer  than  when  it  has  passed  and  is  receding. 

When  the  observer  is  in  motion  toward  the  source,  he  re- 
ceives more  waves  in  each  second  than  when  he  is  at  rest.  The 
additional  waves  received  are  those  which  occupy  the  distance,  v, 
which  he  traverses  in  a  second,  and,  if  X  is  the  wave  length, 
these  are  v/X  in  number.  If  V  be  the  velocity  of  sound  and  n 
the  frequency  of  the  source,  V  =  n\.  Hence  the  increase  in 

frequency  heard  is  ^n   and   the   pitch   as  heard  is  therefore 

(nl+~\ .    For  the  case  in  which  the  hearer  is  in  motion  away 
from  the  source  the  sign  of  v  must  be  reversed. 


MUSICAL  SOUNDS  531 

When  the  source  is  in  motion  toward  the  hearer  the  effect  is 
a  shortening  of  the  wave  length,  for  the  source  is  following  after 
the  approaching  waves,  and  the  crests  therefore  come  closer 
together.  If  the  frequency  of  the  source  is  n  and  its  velocity  is 
v,  during  each  vibration  it  travels  a  distance  v/n  and  each  wave 
length  is  shortened  by  this  amount.  Hence  the  wave  length  of 

V  fV     v\ 

the  sound  heard  is  not  X  =  —  but  X'=  ( )  and  the  frequency 

n  \n     nl 

V        V 
of  the  note  heard  is  —,  —  ~ n.     If  the  source  is  receding  the  sign 

A         V  ~~~  V 

of  v  must  be  reversed. 

The  above  two  expressions  do  not  differ  appreciably  if  v  is 
small  compared  with  V. 

If  a  vibrating  tuning  fork  on  its  sounding  box  be  moved 
rapidly  toward  a  wall  or  blackboard,  an  observer  will  hear  two 
notes  of  different  pitch.  One  is  the  note  heard  directly  from  the 
receding  tuning  fork  and  is  lowered  in  pitch  by  the  motion. 
The  other  note  is  due  to  the  waves  reflected  from  the  wall  and 
this  is  raised  in  pitch.  The  interference  of  these  two  notes 
produces  beats. 

697.  Scale  of  Musical  Sounds. — The  difference  of  pitch  of  two 
sounds  is  called  the  interval  between  them.  In  the  practice  of 
music  these  intervals  are  learned  by  ear,  but  in  scientific  work  an 
interval  is  stated  by  giving  the  ratio  of  the  frequency  of  the 
higher  sound  to  that  of  the  lower.  Certain  intervals  have  re- 
ceived particular  names.  Thus  the  interval  between  two  sounds 
whose  frequencies  are  as  2  : 1  is  called  an  interval  of  an  octave. 

A  certain  succession  of  notes  intermediate  between  a  note 
and  its  octave  has  been  found  suitable  for  musical  purposes  and 
is  called  a  musical  scale  (major  diatonic  scale).  These  notes  are 
called  by  the  letters  of  the  alphabet  from  A  to  G.  Notes  which 
are  each  an  octave  above  one  of  these  are  called  by  letters  from 
a  to  g,  those  still  higher  are  called  a!  to  gf,  a"  to  g",  etc. 


9/8    10/9  16/15  9/8   10/9   9/8  16/15 

c 
24  : 
261 
256 

u 

27 
294 

:  30  :  32 
316  348 
320 

y 
:  36 
391 
384 

u 

:  40 
435 

u    c 

:   45  :  48 
489  522 
512 

In  the  above  the  second  line  contains  the  letters  for  these  notes 


532  SOUND 

(in  the  scale  of  G  major)  and  in  the  first  line  the  fractions  that 
measure  the  intervals  between  consecutive  notes  are  given. 
In  the  third  line  is  a  series  of  numbers  such  that  the  ratio  of  any 
two  is  the  number  that  measures  the  interval  between  the 
corresponding  notes.  These  we  shall  call  the  "proportional 
numbers"  of  the  scale.  The  next  line  contains  the  actual  fre- 
quencies of  these  notes  according  to  the  most  common  method 
of  tuning  orchestral  instruments  ("International"  or  Low  Pitch). 
These  numbers  actually  vary  slightly  in  different  orchestras  but 
their  ratios  remain  the  same.  It  may  be  noted  that  these 
numbers  are  equal  to  the  proportional  numbers  multiplied  by 
10 J.  The  last  line  contains  the  frequencies  of  these  notes  as 
it  has  been  for  a  long  time  the  custom  to  use  them  in  scientific 
work.  These  are  equal  to  the  proportional  numbers  multiplied 
by  lOf . 

Some  other  intervals  in  the  scale  are  named  from  the  order  of 
the  notes  reckoned  from  the  first.  Thus  the  interval  from  the 
first  to  the  fifth,  that  is,  from  C  to  G,  is  called  an  interval  of  a 
fifth.  From  the  proportional  numbers  we  find  that  an  interval 
of  a  fifth  is  equal  to  36/24  or  3/2.  It  will  readily  be  seen  that 
the  intervals  from  E  to  B  and  from  F  to  c  are  also  intervals  of  a 
fifth  (3/2).  The  interval  C  to  F  is  called  an  interval  of  a  fourth 
and  is  equal  to  4/3,  as  are  also  D-G,  E-A,  G-c.  The  intervals 
C-E,  F-A,  G-B  are  intervals  of  a  third  equal  to  5/4.  C-D, 
D-E,  G-A,  A-B  are  all  called  whole  tone  intervals,  for  while 
they  differ  slightly  the  difference  is  only  that  between  10/9 
and  9/8  and  can  hardly  be  detected  by  most  ears.  The  intervals 
E-F  and  B-C  are  much  smaller  and  are  called  semi-tone  intervals. 
For  musical  purposes  intermediate  notes  named  from  these  same 
letters  but  "sharped"  or  "flatted"  (the  black  keys  on  a  piano) 
are  used,  but  further  information  on  this  subject  belongs  to  the 
study  of  music. 

698.  Quality. — Two  musical  sounds  of  the  same  pitch  and 
loudness  may  seem  to  the  ear  to  be  quite  different.  If  the  note 
C  be  sounded  on  a  violin  and  on  an  organ,  and  be  also  sung,  the 
ear  will  recognize  the  source  at  once.  This  difference  we  call  a 
difference  in  quality. 

On  what  does  quality  depend?  We  had  no  difficulty  in 
connecting  loudness  with  the  amplitude  of  the  wave-motion  and 


MUSICAL  SOUNDS 


533 


pitch  with  the  wave  length.  Evidently  quality  must  depend  on 
the  only  other  property  in  which  air-waves  differ,  namely,  the 
wave-form  as  shown  by  curves  representing  the  waves.  Fig.  465 
shows  three  waves  of  the  same  amplitude  and  length  but  of 
different  forms.  The  upper  is  what  we  have  called  a  simple 
harmonic  wave  and  is  the  form  of  wave  given  off  by  a  tuning- 
fork  or  an  organ-pipe  when 
sounded  very  softly.  The 
second  is  more  like  the  wave 
from  a  violin  string  and  the 
third  like  the  wave  from  the 
human  voice  when  singing 

the  vowel  ah  (Miller).     But          f\      f\         f\      f\          ( 
voices  differ  greatly,  and  two      ~J  \j^    \rJ  [f^     \^J 

Voices  Singing  the  Same  note  Fla.  465.— Different  forms  of  waves. 

produce  waves  of  somewhat 

different  form.     The  same  is  true  of  waves  emitted  by  a  violin. 

699.  Harmonics. — Notes  with  frequencies  2,  3,  4,  etc.  times 
the  frequency  of  another  note  are  called  harmonics  of  the  latter. 
Thus  if  the  frequency  of  C  is  256,  the  first  harmonic  of  C  is  c 
which  has  a  frequency  of  512  and  is  an  octave  above  C.  The 
second  harmonic  of  C  is  a  note  with  a  frequency  of  768  and  since 
768/512  is  the  same  as  3/2,  the  second  harmonic  of  C  is  a  fifth 
above  c,  that  is,  it  is  g.  The  third  harmonic  of  C  has  a  frequency 
of  1024  and  since  1024/512  is  the  same  as  2/1,  it  is  an  octave  above 
c  or  two  octaves  above  C.  Hence  it  is  c'. 

Briefly  stated,  notes  whose  frequencies  are  2N,  3N,  4W,  etc., 
are  called  the  harmonics  of  the  note  whose  frequency  is  N  and 
the  latter  is  called  the  fundamental  of  the  harmonics. 

If  we  now  sound  together  such  a  series  of  notes,  either  with 
tuning-forks  or  on  a  piano,  it  will  be  found  that  (up  to  the  seventh 
in  the  series)  they  combine  together  so  harmoniously  that  it  may 
be  difficult  to  hear  them  separately.  Thus  the  term  harmonic 
is  very  appropriate. 

600.  Overtones. — If  the  sound  from  a  large  bell  be  listened  to 
attentively,  it  will  be  found  that  it  consists  of  several  sounds  of 
different  pitch.  The  deepest  of  these  is  called  the  fundamental 
of  the  bell  and  the  others  are  called  overtones  of  the  bell.  The 
intervals  between  these  overtones  may  be  quite  different  on 


534  SOUND 

different  bells.  In  some  cases  the  combination  is  such  that  the 
sound  of  the  bell  is  pleasing.  Other  bells  give  an  unpleasing  or 
harsh  combination.  A  note  produced  on  a  violin  by  a  beginner 
is  apt  to  consist  of  an  unpleasant  combination  of  a  fundamental 
and  overtones,  while  an  accomplished  violinist  has  learned  how 
to  produce  an  agreeable  combination. 

The  difference  between  the  terms  harmonic  and  overtone  should 
be  noted.  Harmonics  refer  to  notes  however  they  may  be  pro- 
duced. When  we  are  given  the  frequency  of  a  fundamental  we 
can  at  once  calculate  the  frequencies  of  its  harmonics  by  multi- 
plying by  2, 3, 4,  etc.  Overtones  refer  to  a  particular  instrument. 
An  overtone  of  an  instrument  may  not  be  a  harmonic  of  the  funda- 
mental, but  we  shall  see  that  the  overtones  of  some  instruments 
are  harmonics  of  their  fundamental  tones. 

601.  Elementary  and  Compound  Sounds. — Most  people  can 
without  any  special  training  distinguish  a  single  sound  from  a 
mixture  of  sounds.  Thus  a  tuning  fork  emits  a  single  sound 
when  it  is  struck  gently.  When  it  is  struck  violently,  or  when  two 
tuning  forks  are  sounded  at  the  same  time,  the  ear  can  usually 
tell  us  that  a  mixture  of  sounds  is  heard. 

A  well-trained  ear  is  capable  of  doing  more  than  this.  For 
example,  it  can  detect  that,  when  a  single  piano  string  is  struck, 
the  sound  produced  is  not  single  but  consists  of  a  fundamental 
and  various  harmonics.  Thus  the  ear  can  do  for  musical  sounds 
what  the  chemist  can  do  for  chemical  compounds,  namely, 
resolve  them  into  their  elements.  The  simple  or  elementary 
sounds  of  music  are  those  in  which  the  ear  can  detect  no  mixture 
of  sounds.  These  are  called  pure  tones.  This  at  once  suggests 
another  question.  What  is  the  difference  of  the  wave  form  of  a 
pure  tone  and  that  of  a  mixed  note,  consisting  of  a  fundamental 
and  harmonics? 

Now  we  have  already  seen  that  there  are  means  by  which  a 
vibrating  body  can  be  made  to  record  the  form  of  its  vibrations 
(§595).  By  these  it  is  found  that  the  vibrations  of  a  body  emitting 
a  pure  tone  are  simple  harmonic  and  the  sound  waves  it  emits  are 
simple  harmonic  waves.  We  have  also  seen  that  simple  harmonic 
waves  of  different  length  when  added  together  produce  complex 
waves  which  may  differ  markedly  from  the  simple  harmonic  form. 
Thus  harmonics  present  with  a  fundamental  alter  the  form  of  the 


MUSICAL  SOUNDS  535 

wave,  and  it  is  this  wave  form1  that  determines  the  quality  of  the 
sound  heard. 

602.  Beats  Between  Sounds. — When  a  white  key  and  an  adja- 
cent black  key  near  the  bottom  of  the  keyboard  of  a  piano  are 
struck  at  the  same  time,  a  distinct  throbbing  of  the  sound  can 
be  heard.     The  throbs  are  slow  and  can  almost  be  counted. 
When  the  same  is  tried  higher  up  on  the  keyboard,  the  throbs  are 
more  rapid.     Throbs  produced  by  two  sounds  of  nearly  the  same 
pitch  are  called  beats. 

If  two  tuning  forks  of  nearly  the  same  pitch  and  mounted  on 
their  sound  boxes  be  thrown  slightly  out  of  unison  by  attaching 
a  small  piece  of  wax  to  a  prong  of  one,  and  if  they  be  then 
sounded  strongly  by  a  bow,  beats  slow  enough  to  be  counted 
will  be  heard.  When  a  larger  piece  of  wax  is  used  the  difference 
of  pitch  of  the  forks  is  increased  and  the  frequency  of  the  beats 
also  increases.  In  fact  in  all  cases,  the  frequency  of  the  beats 
between  two  notes  is  equal  to  the  difference  of  the  frequencies  of  the 
notes.  - 

Since  beats  are  due  to  two  wave  train.s,  of  different  wave 
lengths,  coming  to  the  ear  at  the  same  time,  they  are  the  result 
of  what  we  have  called  interference  of  waves  (§247).  Fig.  152 
may  be  taken  as  representing  the  production  of  beats  between 
two  trains  of  sound  waves. 

603.  Nature  of  Vocal  Sounds. — A  vowel  is  a  sustained  sound  of  a  variable 
pitch  produced  by  holding  the  vocal  cords  and  the  resonating  cavities  of  the 
niouth  and  throat  in  definite  configurations,  while  consonants  are  explosive 
noises  produced  by  the  changes  in  the  vocal  organs  preceding  or  following  a 
vowel  sound.     Vowels  uttered  by  the  same  voice  at  the  same  pitch  differ 
in  quality.    Fig.  465  shows  a  record  obtained  by  Miller  for  the  vowel  ah  in 
father  at  a  pitch  of  182  by  means  of  his  phonodeik  (§587).     The  precise 
cause  of  the  difference  of  quality  of  different  vowels  has  been  a  matter  of 
long  dispute.     The  first  and  natural  supposition  was  that  a  certain  vowel 
consisted  of  a  definite  combination  of  a  fundamental  and  its  harmonics  in 
fixed  proportions,  whatever  the  pitch  of  the  fundamental  might  be.     If  so 
the  curve  of  Fig.  465  would  be  of  the  same  form  (for  the  same  voice)  whatever 
the  pitch  of  the  fundamental  might  be.     This  is  contrary  to  results  obtained 
by  Miller  and  others.     Helmholtz  maintained  that  in  a  particular  vowel 
sound  there  was  a  definite  overtone  characteristic  of  the  vowel  and  having 
the  same  frequency  no  matter  what  the  pitch  of  the  fundamental  might  be. 
Miller  found  that  at  whatever  pitch  (between  129  and  259)  the  vowel  ah 
was  sounded,  60  per  cent,  of  the  energy  was  concentrated  in  an  overtone  of 
about  920  which  was  nearly  but  not  quite  constant.     When  a  phonograph 


536  SOUND 

is  driven  at  less  than  its  normal  speed,  the  quality  of  a  vowel  in  a  singer's 
record  is  altered.  This  is  contrary  to  the  first  view,  since  the  change  of 
speed  does  not  change  the  relative  pitch  and  energy  of  the  combination  of 
fundamental  and  overtones. 

604.  Difference  Tones. — When  beats  between  two  tones  are  sufficiently 
rapid  they  coalesce  and  form  a  distinct  beat  tone,  which  may,  however,  be 
very  weak.    The  frequency  of  the  beat  tone  is  the  difference  of  the  fre- 
quencies of  the  separate  tones  and  when  it  is  distinctly  audible  it  is  usually 
much  lower  than  either  component.     When  careful  attention  is  paid,  the 
beat  tones  between  two  piano  strings  or  two  violin  strings  can  be  clearly 
heard  and  faint  though  the  sounds  may  be  they  have  a  distinct  effect  on 
the  musical  quality  for  trained  ears.     Beat  tones  may,  if  of  too  low  a  pitch 
to  awake  resonance  in  the  instrument,  which  is  the  case  when  the  beating 
tones  are  on  the  lower  strings  of  a  violin,  be  an  effect  purely  in  the  ear,  while 
the  instrument  itself  may  resonate  to  higher  beat  tones. 

SOURCES  OF  MUSICAL  SOUNDS 

605.  Musical  Instruments. — The  number  of  musical  instruments 
is  so  great  that  only  a  few  can  be  referred  to  here.     They  may  be 
most  simply  classified  according  to  the  kind  of  body  that  is 
started  vibrating  in  order  to  produce  the  sound. 

Vibrating  cords  are  used  in  violins,  pianos,  harps,  etc.  In  an 
orchestra  these  are  called  stringed  instruments. 

Vibrating  columns  of  air  are  used  in  organs,  flutes,  clarinets,  etc. 
These  are  called  wind  instruments. 

Vibrating  rods,  plates,  bells,  and  membranes  are  used  in  what  are 
called  percussion  instruments  because  they  are  sounded  by  striking. 

606.  Vibrations  of  Cords. — We  have  already  considered  cases 
of  vibrations  on  cords  when  the  vibrations  are  slow  enough  to 
be  followed  by  the  eye   (Figs.   137,  160).     When  such  waves 
are  continually  reflected  from  both  ends  the  cord  can  divide  up 
into  vibrating  segments  of  stationary  waves.     Each  such  seg- 
ment is  half  of  a  wave-length  in  length.     We  shall  now  consider 
vibrations  on  a  cord  when  they  are  rapid  enough  to  produce 
sound  waves,  but  there  is  no  essential  difference  between  the 
two  cases. 

When  a  cord  stretched  between  two  supports  vibrates,  it  moves 
to  and  fro  between  two  opposite  extreme  positions.  The  forms 
of  the  cord  in  the  two  extreme  positions  can  be  found  by  photo- 
graphy and  in  other  ways  and  they  are  found  to  depend  on  the 
way  in  which  the  cord  is  started.  The  simplest  case  is  when 


SOURCES  OF  MUSICAL  SOUNDS 


537 


the  cord  is  very  gently  bowed  at  the  middle.  In  this  case  the 
form  is  that  of  half  of  a  simple  harmonic  wave,  and,  as  might 
be  expected,  the  note  produced  is  a  pure  or  elementary  tone. 
The  wave-length  is  therefore  21,  where  I  is  the  length  of  the  cord. 
Now  the  velocity  of  waves  equals  the  product  of  frequency  and 
wave-length  or  v  —  n\.  Hence 


and  since  the  value  of  v  is  (§250)\/r/m,  T  being  the  tension 
and  m  the  mass  of  unit  length  of  the  cord, 


From  this  we  see  that  the  frequency  is  inversely  proportional 
to  the  length.  By  shortening  the  "string"  a  violinist  produces 
a  higher  note. 

The  frequency  is  also  proportional  to  the  square  root  of  the 
tension.  To  tune  a  violin  string  'A 

to  the  proper  pitch  the  tension 
is  increased  or  decreased  by 
turning  a  peg  on  which  one  end 
of  the  string  is  wound. 

Finally  the  frequency  is  pro- 
portional inversely  to  the  square 
root  of  the  mass  of  unit  length. 
A  violin  has  four  strings,  the 
thickest  being  used  for  the 
lowest  notes  and  the  thinnest 
for  the  highest  notes. 

The  vibration  of  a  violin  string  is  maintained  by  the  bow,  which 
alternately  grips  and  slips  on  the  string.  The  sound  is  not 
emitted  directly  by  the  strings  bu  by  the  resonating  body  of  the 
instrument. 

607.  Overtones  of  Cords.  —  A  cord  can  also  divide  into  2, 
3,  4,  etc.,  vibrating  segments,  as  we  have  already  seen  in  Fig. 
137  and  this  is  true  whether  the  vibrations  are  too  rapid  to 
be  followed  by  the  eye  or  not.  To  produce  vibrations  with  two 
segments  on  a  cord  touch  it  lightly  at  the  center  and  bow  one-half 
(Fig.  466,  B)  .  It  will  be  seen  that  both  halves  vibrate  and  the  ear 


Fio.  466. 


538  SOUND 

will  hear  a  note  higher  by  an  octave  than  the  fundamental. 
This  is  the  first  harmonic  of  the  fundamental.  If  the  finger  be 
one-third  of  the  length  of  the  string  from  one  end,  and  either 
the  shorter  or  the  longer  part  of  the  string  be  bowed,  the  note 
will  be  the  second  harmonic  of  the  fundamental,  and  so  on.  In 
this  way  the  violinist  can  produce  a  great  variety  of  harmonics  on 
the  various  strings. 

Thus  the  overtones  of  a  cord  are  harmonics  of  the  fundamental. 
To  explain  why  this  should  be  so  let  us  suppose  that  in  any  case 
the  number  of  segments  is  N.  Then  if  I  is  the  length  of  the  cord, 
the  length  of  each  segment  is  I/  N  and  the  wave-length  is  therefore 
From  this  and  the  general  relation  v  =  ri\,  we  get 

v     vN 


Now  I  is  the  constant  length  of  the  cord  and  v  is  also  constant 
so  long  as  th6  tension  of  the  cord  is  unchanged.  Hence  if  we  give 
values  1,  2,  3,  to  N  we  see  that  the  frequencies  are  as  1,  2,  3,  etc. 

608.  Complex  Vibrations  of  Cords.  —  While  we  have  described 
separately  the  different  ways  in  which  a  cord  can  vibrate  they 
can  in  reality  take  place  at  the  same  time.  When  a  cord  is  bowed 
at  one-fourth  of  its  length  from  one  end,  a  good  musical  ear  can 
detect  both  the  fundamental  and  the  first  harmonic.  Or  if, 
while  it  is  vibrating  in  this  condition,  it  be  lightly  touched  at  the 
middle,  the  fundamental  will  cease,  but  the  first  harmonic  will 
be  heard  to  continue  for  a  moment.  In  a  similar  way  we  can 
show  that  other  harmonics  are  present. 

The  particular  overtones  that  are  present  in  the  complex 
vibrations  of  a  cord  depend  chiefly  on  where  it  has  been  struck  or 
bowed,  for  it  is  evident  that  no  form  of  vibration  that  would 
require  the  point  struck  to  be  a  node  can  be  present.  Some  of 
the  possible  combinations  are  more  pleasing  than  others.  The 
most  pleasing  combination  seems  to  be  when  the  cord  is  struck 
at  about  one-eighth  from  one  end  and  this  is  the  actual  practice 
in  the  piano  and,  to  some  extent,  in  the  bowing  of  the  violin 
also. 

The  explanation  of  the  coexistence  of  different  forms  of  vibra- 
tion depends  on  the  fact  that  waves  of  different  length  can  travel 
along  the  cord  at  the  same  time.  Each  such  train,  being  reflected 


SOURCES  OF  MUSICAL  SOUNDS 


539 


at  the  ends,  produces  its  own  system  of  nodes  and  vibrating 
segments. 

The  form  of  the  resultant  wave  on  the  string  evidently  depends 
on  the  particular  combination  of  vibrations  present,  and  for 
each  combination  there  is  a  distinct  characteristic  quality  of  the 
complex  sound  heard.  This  is  another  proof  that  the  quality  of 
a  sound  depends  on  the  form  of  the  wave  that  causes  it. 

609.  Vibrations  of  Air  Columns. — Air  in  a  tube  open  at  one  or 
both  ends  can  be  made  to  vibrate  and  emit  a  musical  sound  by 
blowing  across  an  open  end.  Blowing  with  different  degrees 
of  strength  will  produce  notes  of  different  pitch.  Gentle  blowing 
will  produce  the  lowest  note  and  we  shall  consider^this  first. 

Let  us  first  suppose  that  the  tube  is  closed  at  one  end  and  for 
brevity  let  us  call  this  a  stopped  tube.  The  vibrations  of  the  air 
column  are  stationary  waves  (Fig.  467,a). 
Compressions  started  by  the  blowing  at 
the  open  end  travel  to  the  closed  end  and 
are  there  reflected,  so  that  the  motion  at 
any  point  in  the  tube  is  due  to  the  super- 
position of  two  trains  of  waves  travelling  in 
opposite  directions.  The  closed  end  is  a 
place  of  no  motion  and  is  therefore  a  node 
of  the  stationary  waves.  The  open  end  is 
a  place  of  the  greatest  freedom  of  motion 
and  is  therefore  the  middle  of  a  loop  or 

vibrating  segment.  Now  the  distance  from  a  node  to  the  mid- 
dle of  a  vibrating  segment  is  one-fourth  of  a  wave-length,  X. 
Hence,  if  I  be  the  length  of  the  tube, 

JX  =  Z         or         X  =  4Z 

If  the  tube  is  open  at  both  ends,  it  is  found  that  there  is  a 
node  at  the  middle  (Fig.  467d),  while  the  ends  are  middles  of 
loops.  Hence  it  is  readily  seen  that 

i\  =  }Z        or        \  =  2Z 

From  this  we  see  that  a  stopped  tube  and  an  open  tube  of 
the  same  length  produce  waves  with  lengths  as  2  to  1  and  the 
notes  have  therefore  frequencies  as  1  to  2,  since  frequency  varies 
inversely  as  wave-length.  The  note  of  the  stopped  tube  is 
therefore  an  octave  below  that  of  the  open  tube. 


Tl 

T\ 

i 

I 

A 

A 

X 

y 

A 

\ 

>j 

«      i 

i 

c 

c 

1 

e 

) 

Fia.  467. 

540 


SOUND 


Strictly  speaking  an  open  end  is  not  exactly  the  middle  of  a 
loop,  for  a  condensation  returning  to  the  open  end  does  not  reach 
full  freedom  of  expansion  until  it  has  passed  out  a  short  distance. 
It  has  been  found  that  the  middle  of  the  loop  is  in  reality  about 
.6  of  the  radius  of  the  tube  beyond  the  open  end. 
610.  Resonance  of  Air  Columns. — If  over  the 
open  end  of  a  stopped  tube  we  bring  a  vibrat- 
ing tuning  fork  of  the  same  frequency  as  the 
lowest  note  which  the  tube  will  emit,  resonance, 
or  response  of  the  tube  to  the  fork,  will  take 
place  and  the  note  of  the  fork  will  come  out 
strongly.  A  tube  of  variable  length  can  be  tuned 
exactly  to  the  pitch  of  the  fork  and  for  this  pur- 
pose a  tube  closed  at  the  lower  end  by  a  column 
of  water  that  can  be  raised  or  lowered  is  suit- 
able.  An  open  tube  with  an  extension  piece 
fl  I  that  slips  over  the  end  of  the  tube  can  be  tuned 
v^x  to  a  fork.  If  a  stopped  tube  and  an  open  tube 
be  tuned  to  the  same  fork,  the  former  will  be 
half  as  long  as  the  latter. 
The  interaction  between  the  fork  and  the  tube  is  similar  to 
that  between  the  hand  and  a  swing  when  the  latter  is  being 
started.  The  forward  and  backward  motions  of  the  hand  must 
be  timed  to  agree  with  the  corresponding  natural  motions  of  the 
swing.  Similarly  the  fork  must  complete  one  vibration  in  the 


Fia.  468. — Resonat- 
ing tube. 


Fia.  469. — Spherical  resonator.  FIG.  470. — Tuning-fork  on  resonating  box 

time  of  one  vibration  of  the  air  column,  that  is,  in  the  time  in 
which  the  sound  travels  one  wave  length,  which,  as  we  have  seen 
is  four  times  the  length  of  the  stopped  tube  and  twice  that  of  the 
open  tube. 

Other  forms  of  air  cavity  can  also  resonate  and  emit  definite 


SOURCES  OF  MUSICAL  SOUNDS 


541 


notes.  Spherical  resonators  have  been  much  used  in  analyzing 
complex  sounds.  For  many  purposes  it  is  convenient  to  mount 
heavy  tuning  forks  on  resonating  boxes,  each  of  the  same 
fundamental  pitch  as  the  fork.  While  the  fork  and  the  box  agree 
in  their  fundamental  tones,  they  differ  as  regards  their  over- 
tones and  the  note  emitted  is  much  louder  and  purer  than 
the  fork  alone  can  emit. 

611.  Organ  Pipes.  —  An  organ  pipe  is  a  tube  the  air  in  which 
is  maintained  in  vibration  by  a  jet  of  air  from  a  wind  chest. 
The  jet  is  forced  through  a  narrow  slit  at  the  mouth  of  the  pipe 
and  strikes  against  a  sharp  edge  which  borders  an  opening  on  one 
side  of  the  pipe.  The  farther  end  of  the  pipe  may  be  open  or 
stopped,  but  the  end  at  the  mouth  must  be  regarded  as 
open.  The  first  effect  of  the  jet  is  to  start  either  a 
condensation  or  a  rarefaction  in  the  pipe,  depending  on 
whether  the  jet  is  on  the  whole  directed  more 
to  the  inner  or  the  outer  side  of  the  sharp  edge. 
Thereafter  the  course  of  events  is  determined 
by  the  return  of  the  pulse  after  reflection 
from  the  farther  end.  The  arrival  of  a  con- 
densation forces  the  jet  outward  and  it  then 
by  its  suction  reinforces  the  rarefaction 
which  succeeds  the  condensation  and  travels 
up  the  pipe.  When  a  rarefaction  arrives  the  jet  is 
forced  inward  and  reinforces  the  condensation  which 
follows.  Thus  the  pitch  is  controlled  by  the  natural 
period  of  the  pipe  and  the  energy  required  to  sustain 
the  vibrations  is  derived  from  that  of  the  jet. 

A  stopped  pipe  is  tuned  by  adjusting  a  movable 
plug  which  closes  the  stopped  end.     An  open  pipe  - 
usually  has  a  small  hole  near  the  open  end,  and 
the  pipe   is   tuned   by  adjusting  a  small  strip  of 
metal  that  partly  closes  the  hole. 

An  interesting  illustration  of  another  method  of  supplying  the 
energy  of  vibration  of  a  sounding  tube  is  shown  in  Fig.  472.  A 
long  thick  metal  tube  has  a  sheet  of  wire  gauze  inserted  to  about 
one-fifth  of  its  length  from  one  end.  If  the  gauze  be  heated  by  a 
Bunsen  burner  (with  a  long  extension  tube),  the  tube  will  sound 
very  loudly  when  the  burner  is  removed.  The  whole  explanation 


471. 


Fia.  472. 


542  SOUND 

is  complex,  but  the  supply  of  energy  is  readily  explained.  A 
condensation  coming  to  the  heated  gauze  removes  heat  more 
rapidly  than  a  rarefaction  does.  Hence  the  energy  of  each  con- 
densation is  reinforced  by  the  supply  of  energy  received  from  the 
heated  gauze. 

612.  Tones  of  a  Stopped  Organ  Pipe.— The  closed  end  of  a 
stopped  pipe  is  always  a  node  and  for  the  fundamental  tone  it  is 
the  only  node,  the  open  end  being  the  middle  of  a  loop.     The  first 
overtone  has  a  second  node,  the  open  end  being  still  the  middle 
of  a  loop  (Fig.  467, 6).     Hence  the  second  node  is  one-third  of  the 
length  of  the  pipe  from  the  open  end.     Thus  the  wave  length, 
which  is  always  four  times  the  distance  from  the  middle  of  a  loop 
to  the  nearest  node,  is  4/3  of  the  length  of  the  pipe.     For  the 
second  overtone  there  are  three  nodes  and  so  on.     Thus  it  is 
readily  seen  that  if  I  be  the  length  of  the  pipe  and  N  the  frequency 
of  the  fundamental  the  series  of  wave-lengths  and  notes  are 

41,         41/3,         4Z/5,   .    .    . 

N,        3N,  5N,  .    .    . 

It  will  be  noticed  that  the  overtones  are  harmonics  of  the  funda- 
mental, but  those  of  frequency  2N,  4N,  etc.,  are  not  produced  by 
a  stopped  pipe. 

Except  when  blown  very  softly  a  pipe  produces  one  or  more 
overtones  along  with  the  fundamental  and  the  number  and 
intensity  of  these  determine  the  quality  of  the  complex  sound. 
The  absence  of  the  harmonic  2N  gives  the  stopped  pipe  its 
characteristic  somewhat  dull  tone. 

613.  Tones  of  an  Open  Organ  Pipe. — Both  ends  of  an  open 
pipe  are  mid-points  of  loops.     At  least  one  node  is  required  for 
stationary  waves,  and,  when  an  open  pipe  sounds  its  fundamental 
tone,  this  node  must  evidently  be  at  the  middle  of  the  pipe.     For 
the  first  overtone  there  are  two  nodes,  one  at  one-fourth  of  the 
length  of  the  pipe  from  one  end  and  the  other  at  an  equal  distance 
from  the  other  end  (Fig.  467,  e).     For  the  second  overtone  there 
are  three  nodes  and  so  on.     Thus  it  is  readily  seen  that  the  wave- 
lengths and  frequencies  are 

21,        2112,        21/3,  .    .    . 
N,        2N,          3AT,  . 


SOURCES  OF  MUSICAL  SOUNDS 


543 


The  brighter  quality  of  the  open  pipe  is  due  to  the  presence  of 
the  overtone  2N. 

614.  Longitudinal  Vibrations  of  Rods. — A  rod  clamped  at  its 
middle  point  can  be  made  to  vibrate  by  stroking  it  with  a  rosined 
glove.  Since  both  ends  are  free 
and  the  middle  is  fixed,  the 
fundamental  vibrations  are  similar 
to  those  of  an  open  organ  pipe, 
and  the  wave-length  is  twice  the 
length  of  the  rod.  The  overtones 
follow  the  same  law  as  those  of 
open  organ  pipe.  The  first 


an 


A 

\           /  \    /       \ 
\        /     \/         \ 

\      i      «i  \j 

\        '                 ;  > 

\    /            ;  t 

If          '   \ 

1  '         1    , 

\i       1  I 

;  ! 

\    ' 

B 

a    1      6    y      c 
!               i 

'' 

yyyyyysssx  sssssS&SSSS 


FIQ.  473. 


overtone  may  be  produced  by 
clamping  the  rod  at  points  one- 
fourth  of  the  length  from  the  ends  (as  in  Fig.  476)  and  so  on. 


615.  Transverse  Vibrations  of  Rods. — A  rod  or  strip  of  metal  clamped  at 
one  end  can  be  readily  set  into  transverse  vibrations  by  a  blow  or  push  at 
the  free  end.  Of  two  rods  of  the  same  kind  the  shorter  vibrates  more 
rapidly  than  the  longer,  and  if  the  vibrations  are  rapid  enough  they  produce 
a  musical  note.  The  fixed  end  is,  of  course,  a  node,  but  overtones  with 


FIG.  474. 

two  or  more  nodes  can  also  be  produced.  The  frequencies  of  these  are  not 
in  simple  ratios  to  the  frequency  of  the  fundamental.  Hence  the  overtones 
are  not  harmonics.  Such  vibrations  of  strips  of  wood  or  metal  are  used  as 
reeds  in  many  musical  instruments,  such  as  the  clarinet,  oboe,  bassoon,  and 
reed  organ.  The  reed  is  placed  at  the  end  of  a  pipe  that  resonates  to  the 
vibrations  of  the  reed. 


544  SOUND 

A  rod  clamped  at  the  middle  will  vibrate  transversely  with  a  node  at  that 
point.  A  tuning  fork  consists  essentially  of  such  a  rod  with  the  ends  turned 
parallel.  The  overtones  of  a  tuning  fork  are  not  harmonics  of  the  funda- 
mental, but  if  the  fork  is  sufficiently  thick  and  is  not  struck  too  violently, 
the  overtones,  besides  being  very  high  are  so  weak  as  not  to  produce  dis- 
agreeable effects. 

Large  metal  tubes  vibrating  transversely  are  sometimes  used  in  operatic 
music  to  imitate  deep-toned  bells. 

616.  Vibrations  of  Plates. — A  square  or  round  plate  of  metal, 
the  middle  of  which  is  screwed  to  the  end  of  a  rod,  can  be  made 

to  vibrate  in  a  great  variety  of  forms 
by  stroking  with  a  bow.  The  nodes 
or  lines  of  no  motion  are  beautifully 
shown  by  fine  sand  on  the  plate.  In 
the  fundamental  mode  of  vibration 
there  are  only  two  of  these  lines,  but 
in  the  higher  modes  of  vibration  a  large 
number,  straight  or  curved,  appear. 
Fia  475  617.  Bells. — A  bell  may  be  thought 

of  as  a  round  plate  turned  up  to  a 

cup  shape.  The  fundamental  mode  of  vibration  is  like  that 
of  the  plate,  with  two  nodal  lines;  but  when  the  bell  is  struck, 
many  overtones  are  pruduced  along  with  the  fundamental,  and 
some  of  these  are  so  much  stronger  than  the  fundamental  that 
the  latter  frequently  cannot  be  heard.  In  fact  it  is  usually  the 
fourth  overtone  that  is  loudest  and  is  taken  by  the  ear  as  giving 
the  pitch  of  the  bell. 


VELOCITY  OF  SOUND.     EXPERIMENTAL  METHODS 

618.  Resonating  Tube  Method.— From  the  length  of  a  tube  that 
resonates  to  a  tuning  fork  of  known  frequency  the  velocity  of 
sound  in  the  gas  in  the  tube  can  be  found  by  means  of  the  relation 
V  =  nX.  For  this  purpose  a  glass  tube  open  at  one  end  and  con- 
taining an  adjustable  column  of  water  is  convenient.  From  the 
account  of  the  vibrations  of  a  stopped  organ  pipe  (§  612)  it  is 
seen  that,  if  the  tube  be  of  sufficient  length,  it  can  resonate  when 
the  length  of  the  column  of  gas  is  X/4,  or  3X/4  or  5X/4,  etc.,  where 
X  is  the  wave  length  of  the  sound  of  the  tuning  fork.  When  it 
resonates  with  more  than  one  node  the  distance  between  two 


ACOUSTICS  OF  HALLS  545 

consecutive  nodes  is  accurately  X/2,  while  the  distance  from 
the  upper  node  to  the  open  end  is  only  approximately  X/4. 
Hence  the  best  value  of  the  wave-length  and  velocity  is  obtained 
from  the  former. 

619.  Kundt's  Dust  Tube  Method. — This  method  differs  from 
the  preceding  in  that  a  metal  rod  vibrating  longitudinally  is 
used  instead  of  a  tuning  fork  and,  since  the  pitch  is  very  high, 
the  air  column  divides  into  numerous  vibrating  segments. 
These  segments  are  clearly  shown  by  cork  dust  which  gathers 
at  the  nodes.  One  end  of  the  rod  projects  into  the  tube  and  car- 
ries a  light  disk  of  slightly  smaller  diameter  than  that  of  the  tube. 
It  is  convenient  for  stability  to  clamp  the  rod  at  two  points, 
each  at  one-fourth  of  the  length  of  the  rod  from  one  end.  The 


Fid.  476. 

wave-length  of  the  waves  in  the  rod  is  then  equal  to  the  length, 
L,  of  the  rod.  The  wave-length  of  the  sound  in  the  gas  is  twice 
the  distance  between  two  adjacent  dust  heaps  or  21.  Let  V 
be  the  velocity  of  the  waves  in  the  rod  and  v  the  velocity  of 
sound  in  the  gas.  Then  if  n  be  the  frequency  of  the  vibration, 
which  is  the  same  for  both,  V  =  nL  and  v  =  2rd. 
Hence 

v  _2l 
V~"L 

If  we  assume  the  velocity  of  sound  in  air  to  be  known  we  can, 
from  this  relation,  find  the  velocity  of  the  waves  in  the  rod  and 
also  the  velocity  of  sound  in  any  other  gas  introduced  into  the 
tube.  The  method  also  lends  itself  to  the  study  of  the  effect 
of  change  of  temperature  on  the  velocity  of  sound  and  it  has  been 
modified  so  as  to  make  it  suitable  for  finding  the  velocity  of  sound 
in  a  liquid. 

ACOUSTICS  OF  HALLS 

620.  Defects  in  the  Acoustics  of  Halls. — In  some  halls  the 
" hearing"  is  satisfactory,  in  others  unsatisfactory.     A  full  treat- 

35 


546 


SOUND 


Fia.  477A. — (Sabine,  American  Architect.) 


FIQ.  477B.— (Sabine,  American  Architect.) 


ACOUSTICS  OF  HALLS  547 

ment  of  the  subject  is  impossible  here,  but  a  few  points  may  be 
considered  briefly. 

Sound  reaches  an  auditor  in  a  hall  not  only  by  waves  coming 
directly  from  the  source,  but  also  by  waves  reflected  by  walls 
(including  in  this  term  all  reflecting  surfaces)  and  the  paths  usu- 
ally differ  in  length.  The  result  may  be  an  increase  in  loudness 
but  more  or  less  confusion  of  sound  also  follows.  A  speaker 
utters  two  or  three  syllables  per  second  and  a  musical  instrument 
may  emit  several  notes  in  a  second  and  reflection  causes  an  over- 
lapping of  the  separate  sounds.  The  two  chief  defects  resulting 
may  be  classified  as  echo  and  reverberation.  A  separate  echo  is 
due  to  some  particular  reflecting  surface  and  may  be  reduced  by 
alterations  in  the  surface.  Reverberation  consists  in  a  prolonga- 
tion of  the  sound  in  all  parts  of  the  hall  due  to  repeated  reflections 
from  all  walls,  somewhat  as  a  room  (with  white  walls)  may  be 
illuminated  throughout  by  light  entering  by  a  single  small  win- 
dow. Reverberation  is  always  a  drawback  for  speech  but  it 
cannot  be  eliminated  without  reducing  loudness  to  an  undesirable 
extent.  For  music  a  certain  amount  of  reverberation  is  an  ad- 
vantage, but  a  larger  amount  is  detrimental. 

Fig.  477  shows  the  history  of  a  single  wave  of  sound  started 
from  the  stage  of  a  theater  and  reflected  by  various  surfaces.  In 
A  we  have  the  direct  wave  .07  of  a  second  after  it  has  started, 
together  with  several  reflected  waves  that  will  cause  weak  echoes. 
B  shows  a  part  of  the  direct  wave  .07  of  a  second  later  and  a  com- 
plex system  of  reflected  waves.  The  figures  are  from  photographs 
obtained  by  Professor  W.  C.  Sabine,  to  whom  we  owe  most  of 
our  knowledge  of  Architectural  Acoustics,  by  the  method  referred 
to  in  §588,  a  small  model  of  the  theater  being  used  for  the  purpose. 

621.  Remedy  for  Reverberation. — If  the  walls  of  a  hall  did  not 
reflect  sound  there  would  be  no  reverberation.  When  it  is  pres- 
ent to  an  undesirable  extent  its  amount  may  be  decreased  by 
covering  some  part  of  the  walls  with  soft  materials  that  reflect 
little  sound.  It  is  usually  not  a  matter  of  importance  on  what 
wall  or  part  of  a  wall  the  deadening  material  is  placed.  This  is 
due  to  the  fact  that  the  sound  starts  out  from  the  source  in  all 
directions  and  travels  about  1140  ft.  in  a  second.  This  distance 
is  so  great  compared  with  the  dimensions  of  an  ordinary  hall  that 
the  rays  are  in  general  reflected  many  times  in  a  second  and  as 


548  SOUND 

much  approximately  falls  on  each  square  unit  of  area  wherever 
it  may  be  situated.  Yet  there  are  places  in  some  halls  .where 
more  or  less  than  the  average  amount  of  sound  reaches  the  walls 
and  for  such  special  cases  a  special  study  of  the  hall  is  necessary. 
For  the  details  of  this  we  must  refer  to  Sabine's  papers. 

The  reader  may  have  heard  of  cases  in  which  an  attempt 
has  been  made  to  reduce  reverberation  by  stretching  fine  wires 
across  a  hall.  These  and  other  similar  devices  are  entirely 
useless,  as  has  been  shown  by  careful  study  of  the  results. 

622.  Duration  of  Residual  Sound. — Since  reverberation  is  due 
to  the  continuance  of  a  sound  after  it  has  been  produced,  the 
extent  of  the  reverberation  in  a  hall  can  be  estimated  by  measur- 
ing the  length  of  time  a  sound  is  heard  in  the  hall  after  the  source 
has  ceased  emitting  waves.     Sabine  studied  this  question  by 
means  of  an  organ  pipe  having  a  frequency  of  512  and  blown 
under  a  definite  pressure,  together  with  a  chronograph  for  record- 
ing the  length  of  time  the  residual  sound  was  heard.     He  found 
that  the  residual  sound  was  heard  for  a  measurable  number  of 
seconds  and  died  away  according  to  an  exponential  law.     From 
the  data  obtained  it  was  possible  to  calculate  the  time  required 
for  any  specified  decrease  in  intensity,  and  it  was  convenient  to 
define  "the  duration  of  residual  sound "  in  a  hall  as  the  time  re- 
quired for  the  intensity  to  fall  to  one  millionth  (10~6)  of  its  orig- 
inal value.     This  we  shall  denote  by  T.    While  the  value  of  T 
is  different  for  notes  of  different  pitch  and  such  variations  were 
studied  by  Sabine,  it  will  be  convenient  in  this  brief  account  to 
restrict  it  to  the  particular  frequency  (512)  referred  to,  for  it  was 
found  that,  for  orchestral  music,  T,  thus  defined,  should  be  about 
2.3  seconds.    A  hall  for  which  T  is  as  great  as  3  seconds  or  as 
small  as  2  seconds  is  not  satisfactory  for  orchestral  music. 

623.  Coefficients  of  Absorption. — An  open  window  may  be 
regarded  as  a  non-reflector,  that  is  a  perfect  absorber,  since  all 
the  sound  which  falls  on  it  passes  out.     Sabine  found  by  experi- 
ments  on  the   duration   of  residual   sound  in  an   auditorium 
what  area  of  any  particular  drapery  would  cause  the  same  ab- 
sorption of  sound  as  1  sq.  m.  of  open  window,  and  therefrom  he 
deduced  the  absorbing  power  of  the  material.     If,  for  example, 
it  took  3  sq.  m.  of  a  material  to  produce  the  same  decrease 
of  the  residual  sound  as  1  sq.  m.  of  open  window,  each^  square 


ACOUSTICS  OF  HALLS  549 

meter  of  the  material  must  have  absorbed  one-third  as  much 
as  the  square  meter  of  open  window.  But  the  latter  absorbed 
all  of  the  sound  that  fell  on  it.  Hence  the  drapery  absorbed 
one-third  of  the  incident  sound,  therefore  its  coefficient  of  ab- 
sorption was  J.  Sabine  found  by  such  experiments  the  ab- 
sorbing power  of  an  average  auditor  and  of  various  articles 
which  are  commonly  found  in  a  hall.  Some  of  these  coefficients 
are  given  in  the  following  table.  For  additional  data  Sabine's 
papers  must  be  consulted. 

COEFFICIENTS  OF  ABSORPTION 

Open  window 1 .000 

Wood  sheathing  (hard  pine). 061 

Plaster  on  lath 034 

Plaster  on  tile 025 

Brick  set  in  Portland  cement 025 

Carpet  rugs 20 

Shelia  curtains.. 23 

Hair  felt  (2.5  cm.  thick,  8  cm.  from  wall) 78 

Audience  (per  person). 44 

624.  Total  Absorbing  Power  of  a  Hall. — Each  surface  absorbs 
in  proportion  to  its  area  and  coefficient  of  absorption.     If,  there- 
fore, we  multiply  each  area  of  surface  of  a  particular  kind  by  its 
coefficient  of  absorption  and  add  the  products  we  shall  get  the 
total  absorbing  power,  a,  of  the  auditorium,  or 

a  =  aiSi  +  0,282  +    .    .    .    =  Sas 

To  take  account  of  the  audience  in  this  sum  we  multiply 
the  number  supposed  to  be  present  by  the  absorbing  power  per 
person. 

625.  Effect  of  Size  and  Shape  of  Hall. — Sabine  found  that  in 
general  the  shape  of  a  hall  of  given  total  volume  and  given 
total  absorbing  power  had,  to  a  first  approximation  at  least,  no 
effect  on  the  length  of  duration  of  residual  sound  in  the  hall. 
In  fact,  other  things  being  equal,  if  the  sound  diffuses  as  rapidly 
as  we  have  supposed  to  all  parts  of  a  hall  and  so  becomes  nearly 
uniformly  distributed  throughout,  the  absorption  must  depend 
only  on  the  average  frequency  of  reflection  of  the  rays  starting 
from  the  source,  and  it  can  be  shown  theoretically  that  this 
average  frequency  depends  only  on  the  volume. 


550  SOUND 

626.  Sabine's  Formula. — Sabine  found  by  experiments  on  many 
halls  that  the  length  of  duration  of  residual  sound  could  be  ex- 
pressed by 

17 

T  =  .164  - 
a 

where  V  is  the  total  volume  of  the  hall  in  cubic  meters  and  a 
is  its  total  absorbing  power,  areas  being  calculated  in  square 
meters.  Strictly  speaking  the  velocity  of  sound  should  appear 
in  the  denominator  but  it  is  assumed  to  have  the  average  value  of 
342  m/s  and  is  included  in  the  factor  .164. 

The  above  formula  can  be  applied  at  once  to  calculating  the 
duration  of  residual  sound  in  an  auditorium,  existing  or  con- 
templated, and  thus  its  suitability  for  music,  and  approximately 
also  for  speech,  can  be  tested.  It  can  be  applied  also  to  remedy 
the  defects  in  an  existing  hall  by  suitably  varying  the  value  of  a. 
From  the  formula  we  can  also  derive  immediate  answers  to  such 
questions  as  the  effect  of  the  presence  of  the  audience  on  the 
"hearing"  in  a  hall,  the  result  of  copying  an  existing  satisfactory 
hall  but  on  a  larger  scale  and  so  on.  These  may  be  left  as  exer- 
cise to  the  reader. 

References 

RAYLEIGH'S  Theory  of  Sound  is  a  standard  work  containing  both  theory  and 

experiment. 
HELMHOLTZ'B  Sensations  of  Tone  is  a  classical  work  of  great  originality. 

POTNTING  AND  THOMSON'S  Sound. 

POTNTING  on  Sound  in  the  Encyclopedia  Britannica. 
BARTON'S  Sound. 

These  three  works  contain  clear  statements  of  experimental  facts  with 
considerable  theory. 
MILLER'S  Science  of  Musical  Sounds  is  an  interesting  and  original  work, 

copiously  illustrated,  on  the  experimental  analysis  of  sound  waves. 
SABINE'S  articles  on  Architectural  Acoustics  (American  Architect,  1900,  1913; 

Proc.  American  Academy  of  Arts  and  Sciences^  1906)  are  of  great 

originality  and  of  fundamental  importance. 

Problems 

1.  What  is  the  ratio  of  the  velocity  of  sound  on  a  hot  summer  day  (35°C.)  to 
that  on  a  cold  winter  day  (-  20°C.)? 

2.  A  Band  is  playing  on  a  steamer  which  is  travelling  with  a  speed  of  20 


PROBLEMS  551 

miles  per  hour.     With  what  speed  must  an  observer  move  in  the  opposite 
direction  in  order  that  every  note  may  be  lowered  by  a  semitone? 

3.  Find  the  pitch  of  the  fundamental  and  of  the  first  two  overtones  of  a 
stopped  pipe  1  m.  long  at  16°C. 

4.  What  is  the  length  of  an  open  pipe,  if  the  pitch  of  its  first  overtone  is  512? 

5.  A  tuning  fork  makes  2  beats  per  second  with  a  standard  fork  the  frequency 
of  which  is  512.     When  a  small  piece  of  wax  is  placed  on  a  prong  of  the 
former  the  number  of  beats  is  decreased.     What  is  its  pitch? 

\  A  tuning  fork  of  frequency  384  makes  2  beats  per  second  with  a  vibrating 
string.  In  what  proportion  must  the  tension  of  the  string  be  changed 
in  order  that  the  two  may  be  in  unison? 

7.  A  stopped  pipe  resonates  to  a  tuning  fork,  the  pitch  of  which  is  258,  when 
adjusted  to  a  length  of  32  cm.  and  also  when  the  length  is  98  cm.,  the 
temperature  being  18°C.     What  is  the  ratio  of  the  specific  heats  of  air? 

8.  A  glass  rod  is  used  in  Kundt's  experiment.     If  the  dust  piles  are  5  cm. 
apart  and  the  distance  between  the  clamped  points  on  the  rod  is  70  cm., 
what  is  Young's  modulus  for  this  glass,  the  density  of  glass  being  2.7 
gm./cu.  cm.? 


LIGHT. 

BY  E.  PERCIVAL  LEWIS,  PH.  D. 
Professor  of  Physics  in  the  University  of  California. 

GENERAL  PROPERTIES. 

627.  Radiation. — As  in  the  cases  of  Heat  and  Sound,  the  word 
Light  has  acquired  two  distinct  meanings.  The  primary  and 
more  familiar  one  is  that  which  is  associated  with  the  sensation  of 
vision.  Nearly  all  that  relates  to  this  aspect  of  the  subject  lies 
within  the  province  of  the  psychologist.  The  physicist,  however, 
generally  uses  the  term  in  an  objective  sense,  with  reference  to 
the  external  agencies  which  may  excite  the  sensation  of  light 
if  allowed  to  act  on  the  eye.  The  visible  radiation  which  affects 
a  normal  eye  will  also  affect  a  photographic  plate,  a  thermometer, 
or  other  sensitive  detector  of  heat.  It  will  be  found,  after 
analyzing  the  radiation  from  the  sun,  electric  light,  or  other 
sources  with  a  prism,  that  beyond  the  violet  and  the  red  lie  non- 
luminous  radiations  which  will  affect  a  photographic  plate  or  a 
thermometer,  and  it  has  been  shown  that  the  oscillations  of 
an  electric  spark  between  metallic  terminals  are  accompanied  by 
the  radiation  of  electric  waves  through  space.  There  is,  as  we 
shall  see,  no  fundamental  qualitative  difference  between  these 
various  radiations,  and  it  is  due  merely  to  a  special  property  of 
the  eye  that  some  of  them  excite  the  sensation  of  light  while 
'  others  do  not.  This  is  analogous  to  the  selective  resonance  of  a 
piano  wire,  which  will  respond  to  certain  notes  and  not  to  others. 
Just  as  some  ears  can  detect  sounds  of  such  high  pitch  as  to  be 
inaudible  to  others,  some  eyes  can  detect  ether  radiations  lying 
somewhat  beyond  the  limits  of  perception  of  the  ordinary  eye. 
In  the  following  pages  the  whole  range  of  these  radiations  so  far 
as  they  are  known  will  be  considered.  As  a  matter  of  conveni- 
ence, the  term  Light,  which  strictly  speaking  would  apply  only 

553 


554  LIGHT 

to  the  radiations  exciting  the  sensation  of  light,  will  be  used  in  a 
figurative  sense  to  include  the  entire  range  of  radiations  which 
are  alike  in  their  general  properties,  and  which  were  once  very 
artificially  classified  as  luminous,  actinic,  and  heat  radiations. 

628.  Sources  of  Light. — The  best  known  are  the  sun,  the 
physical  nature  and  condition  of  which  are  as  yet  not  fully 
understood,  solid  bodies  at  a  high  temperature,  such  as  the 
calcium  light,  electric  arc  and  incandescent  lights,  and  luminous 
flames.  If  a  piece  of  cold  porcelain  is  held  over  the  flame  of  a 
candle,  lamp,  or  gas  jet,  it  will  become  covered  with  finely 
divided  carbon,  while  no  such  deposit  is  observed  in  the  case  of  a 
non-luminous  Bunsen  or  alcohol  flame.  This  suggests  that  the 
luminosity  of  these  flames  is  due  to  the  presence  of  incandescent 
carbon  particles.  This  idea  is  strengthened  by  the  fact  that 
when  the  base  of  a  Bunsen  burner  is  closed  the  flame  becomes 
luminous  and  smoky;  when  open,  enough  oxygen  is  admitted 
to  combine  with  all  the  carbon  set  free  by  the  dissociation  of  the 
coal  gas,  and  the  flame  is  then  non-luminous.  The  carbon 
oxides  formed  are  permanent  gases,  and  there  is  no  evidence 
that  such  gases  can  be  made  luminous  by  high  temperature  alone. 
Any  gas  may  be  made  luminous,  however,  by  the  passage  of  an 
electric  discharge  through  it,  but  this  luminosity  does  not  seem 
to  be  accompanied  by  very  high  temperature.  There  are,  in 
fact,  many  cases  in  which  light  is  emitted  at  a  very  low  average 
temperature  of  the  source.  As  examples  may  be  mentioned 
the  various  types  of  phosphorescence,  some  of  which  are  most 
active  at  temperatures  as  low  as  that  of  liquid  air,  the  aurora 
due  to  electrical  discharges  through  the  highly  rarefied  and  very 
cold  upper  atmosphere,  and  the  light  emitted  by  fire-flies  and 
glow-worms. 

629.  Rectilinear  Propagation. — One  of  the  earliest  observa- 
tions concerning  light  was  that  it  travels  in  straight  lines,  in  a 
homogeneous  medium.     These  lines  of  propagation  or  "rays" 
may  be  made  to  alter  their  direction  only  by  one  of  two  methods 
— by  reflection,  when  they  fall  on  the  boundary  between  two 
media,  or  by  refraction,  when  they  pass  obliquely  from  one 
medium  to  another,  or  through  a  medium  of  varying  density. 

630.  Shadows  and  Eclipses. — Rays  pass  in  straight  lines  by 
the  edges  of  an  obstacle,  so  that  the  space  behind  it  is  screened 


GENERAL  PROPERTIES  555 

from  the  light.  If  the  latter  comes  from  a  very  small  or  "  point " 
source  the  shadow  would  be  sharply  defined  if  the  propagation 
were  strictly  rectilinear;  as  a  matter  of  fact,  close  observation 
shows  in  all  cases  that  the  light  fades  gradually  into  the  shadow. 
This  very  significant  fact  proves  that  light  travels  only  approxi- 
mately in  straight  lines;  there  is  always  more  or  less  lateral  spread- 
ing. Strictly  speaking,  there  is,  then,  no  such  thing  as  a  ray  of 
light,  if  we  mean  by  this  term  propagation  along  a  geometrical 
line.  The  explanation  of  this  spreading  will  be  given  later 
(§701  etseq.). 

A  more  obvious  cause  of  the  lack  of  sharpness  in  shadows  is  to  be  found 
in  the  fact  that  most  sources  of  light  are  not  even  approximately  points, 
but  are  of  finite  area.     This  gives 
rise   to   the  distribution   of  light 
and  shadow  shown  in  Fig.  478  and 
Fig.  479.     The  first  represents  the 
shadow  cast  by  an  object  larger 
than  the  source;  the  second,  that  Fia.  473. 

due  to  an  object  smaller  than  the 

source;  for  example,  the  shadow  of  the  earth  due  to  the  sun.  In  each  case 
there  is  a  region  of  complete  shadow  behind  the  obstacle,  called  the 
umbra,  into  which  no  light  from  any  part  of  the  source  can  enter.  Around 

this  there  is  a  region  called  the 
penumbra,  which  receives  light 
from  a  part  of  the  source,  the 
effective  portion  of  the  latter  in- 
creasing in  going  outward  from 
the  umbra.  When  the  moon  lies 
FIQ.  479.  entirely  within  the  shadow  cone 

of  the  earth  it  is  said  to  be  com- 
pletely eclipsed ;  when  it  passes  through  the  penumbra  or  partly  through 
the  umbra  and  partly  through  the  penumbra  it  is  partially  eclipsed. 

631.  Parallax. — This  well-known  phenomenon  depends  upon 
the  rectilinear  propagation  of  light.  By  parallax  is  meant  the 
apparent  displacement  of  an  object  due  to  the  real  displacement 
of  the  observer.  For  example,  if  the  observer  moves  from  Ol  to 
02  (Fig.  480)  A  will  appear  to  be  displaced  an  angular  distance 
a-f/?  to  the  left  with  reference  to  B.  That  object  which  seems 
to  be  displaced  in  a  direction  opposite  to  the  motion  of  the 
observer  is  evidently  the  nearer.  To  one  traveling  on  a  railroad 
train  objects  near  at  hand  appear  to  be  moving  backward,  those 


556 


LIGHT 


at  a  distance  in  the  same  direction  as  the  observer.  If  two 
objects  are  coincident  in  position  or  equally  distant  their 
relative  parallax  vanishes.  This  gives  a  useful  method  of 
finding  the  apparent  position  of  the  image  formed  by  a  lens  or 
mirror,  or  of  focusing  the  cross  thread  of  a  telescope.  When 
the  latter  and  the  image  of  a  distant  object 
are  both  distinctly  seen  and  have  no  relative 
parallax  they  are  coincident  in  position  and 
both  in  focus. 

In  astronomy  horizontal  parallax  is  defined  as 
the  angle  subtended  by  the  semi-diameter  of  the 
earth  from  any  body  of  the  solar  system.  Annual 
parallax  is  the  angle  subtended  by  the  semi-diame- 
ter of  the  earth's  orbit  from  the  more  distant 
fixed  stars.  The  distance  between  the  sun  and  the 
earth  may  be  determined  by  observing  the  transit 
of  an  inferior  planet,  Venus  for  example,  across  the 
sun's  disk.  Observers  at  A  and  B  (Fig.  481)  note  the 
instants  at  which  Venus  appears  to  enter  the  sun's 
disk  as  viewed  from  their  respective  stations.  From 
the  interval  between  these  two  contacts  and  the  known  angular  velocity 
of  Venus  around  the  sun  the  angle  a  may  be  determined,  and  from  that 
and  the  base  line  AB  the  horizontal  parallax  and  distance  of  the  sun  may 
be  calculated.  Of  course  correction  must  be  made  for  the  motion  of  the 
earth  between  the  instants  of  contact. 


Fio.   480. 


632.  Pinhole  Image. — Another  effect  of  the  approximately 
rectilinear  propagation  of  light  is  the  formation  of  an  inverted 
image  of  the  source  by  light  passing  through  a  small  orifice  such 
as  a  pinhole.  If  any  source,  for  example,  a  candle,  is  placed 
opposite  such  a  hole  in  a  screen  S^ 
(Fig.  483)  light  from  the  point  P  will 
pass  through  the  opening  in  a  narrow 
cone  or  pencil  and  illuminate  a  small 
patch  at  P2  on  a  screen  S2.  Light 
from  Q  will  form  a  small  patch  at  Q2, 
and  light  from  any  other  point  of  the  flame  will  fall  on  a  corre- 
sponding point  of  the  screen  S2.  The  group  of  patches  will  in 
form,  color,  and  relative  brightness  reproduce  the  candle  flame, 
but  evidently  inverted  in  position.  The  pinhole  forms  an  image 
like  that  due  to  a  condensing  lens,  but  the  total  light  in  the 


Fio.  481. 


GENERAL  PROPERTIES  557 

pinhole  image  will  be  less  than  that  formed  by  the  lens  in  the 
proportion  of  the  area  of  the  pinhole  to  that  of  the  lens.  As  the 
image  is  due  to  a  group  of  overlapping  patches,  it  will  not  be  so 
sharp  in  outline  as  that  made  by  the  lens.  The  blurring  will  in- 
crease with  the  size  of  the  opening  or  when 
the  source  is  brought  near  the  screen, 
thus  increasing  the  angle  of  the  trans- 
mitted cone.  The  object  and  its  image  Q 
subtend  equal  angles  at  the  pinhole,  so 
that  their  linear  magnitudes  are  in  the 
same  ratio  as  their  respective  distances  u 
and  v  from  the  screen  £x.  This  is  also 
true  of  images  formed  by  any  other  optical  device,  such  as  a  mirror 
or  lens.  Landscape  photographs  of  great  softness  and  beauty 
may  be  made  by  the  use  of  the  pinhole  camera. 

633.  Reflection,  Regular  and  Diffuse. — When  light  falls  on  a 
smooth  polished  surface  it  is  reflected  in  a  definite  direction.    This 
is  called  regular  reflection.     The  plane  including  the  direction  of 
the  incident  light  and  the  normal  to  the  surface  at  the  point  of 
incidence  is  called  the  plane  of  incidence.     The  angle  between  the 
incident  pencil  and  the  normal  to  the  surface  is  called  the  angle 
of  incidence;  that  between  the  reflected  pencil  and  the  normal  is 
called  the  angle  of  reflection.     Experiment  shows  that  (1)  the 
angle  of  reflection  is  equal  to  the  angle  of  incidence;  (2)  the  reflected 
pencil  lies  in  the  plane  of  incidence.     It  is  evident  from  the  first 
law  that  if  a  mirror  is  rotated  through  a  given  angle  about  an 
axis  perpendicular  to  the  plane  of  incidence,  the  reflected  pencil 
will  be  rotated  through  twice  that  angle. 

When  light  falls  on  a  rough  unpolished  surface  it  is  reflected 
in  all  directions.  This  is  called  diffuse  or  irregular  reflection. 
There  is  no  essential  difference  between  regular  and  diffuse 
reflection  except  that  in  the  latter  we  may  imagine  reflection  to 
take  place  from  an  infinite  number  of  infinitesimal  plane  surfaces 
orientated  in  all  directions. 

634.  Visibility  of  Objects. — On  a  clear  night,  where  there  is  no 
moonlight,  the  stars  and  planets  appear  against  a  background  of 
black  sky.     The  space  around  the  earth's  shadow  cone  is  filled 
with  sunlight,  but  we  do  not  see  it  unless  it  is  reflected  from 
some  planet  or  the  moon.     If  a  beam  of  light  is  passed  through  a 


558  LIGHT 

vessel  of  distilled  water  its  path  is  invisible.  If  a  beam  of  sunlight 
enters  a  dark  room  it  cannot  be  seen  unless  dust  particles  are 
floating  in  the  air.  A  drop  of  milk  in  the  water  or  a  little  dust 
stirred  up  in  the  room  will  cause  the  path  of  the*  light  to  flash 
out  brilliantly.  Such  experiments  show,  as  might  be  expected, 
that  light  does  not  excite  the  sensation  of  luminosity  unless  it 
enters  the  eye  directly  from  the  source  or  by  reflection.  Ordi- 
nary objects  are  visible  because  they  reflect  light  diffusely  into 
the  eye,  and  they  may  be  regarded  as  secondary  sources  of  radia- 
tion. A  perfect  reflector  would  itself  be  invisible,  all  the  light 
reflected  from  it  appearing  to  come  from  the  image  of  the  source, 
not  from  the  reflector. 

635.  Transmission   and   Absorption. — Light   travels   through 
some  media,  for  example   most   gases,  glass  and  water,  with 
scarcely  any  appreciable  diminution  of  intensity.     Other  media 
may  transmit  little  or  none,  or  certain  colors  only;  such  media  are 
said  to  show  general  or  selective  absorption.     In  cases  where 
absorption  occurs  there  appears  to  be  a  loss  of  radiant  energy, 
but  it  may  be  shown  that  it  changed  to  other  forms,  usually 
heat  (§330  etseq.). 

636.  Transparency,   Translucency,    Opacity. — Any   substance 
which  transmits  a  large  fraction  of  the  incident  light  without 
scattering  it  is  said  to  be  transparent.    As  indicated  by  this  term, 
objects  may  be  seen  clearly  through  such  substances.     Objects 
which  absorb  all  the  unreflected  incident  light  are  said  to  be 
opaque,   and   act   as   perfect   screens.     Evidently   any   perfect 
reflector  must  also  be  perfectly  opaque,  but  in  this  case  opacity 
is  not  due  to  absorption.     Substances  differ  widely  in  these 
properties,  varying  from  almost  perfect  transparency  to  almost 
perfect  opacity.     The  most  transparent  media  known  show  some 
absorption,  which  increases  with  the  length  of  path;  hence  any 
substance  will  become  opaque  if  a  sufficient  thickness  is  taken. 

No  light  penetrates  to  great  depths  in  the  ocean,  although  a  layer  of 
water  of  considerable  thickness  is  transparent.  On  the  other  hand,  light 
will  penetrate  to  a  slight  depth  in  any  medium,  so  that  thin  layers  of  metal 
or  of  carbon  are  found  to  be  transparent.  Some  substances  are  selectively 
transparent ;  red  glass  will  freely  transmit  red  light,  but  not  the  other  colors, 
and  a  thin  sheet  of  hard  rubber,  which  appears  to  be  opaque,  will  transmit 
radiations  lying  a  little  outside  the  red  of  the  spectrum. 

Some  substances  transmit  light,  but  scatter  it  so  that  objects  cannot  te 


GENERAL  PROPERTIES  559 

clearly  seen  through  them.  These  substances  are  called  translucent. 
The  effect  is  caused  by  diffuse  reflection  within  the  medium,  due  to  discon- 
tinuity or  non-homogeneity  of  structure,  as  in  the  case  of  powdered  glass, 
paper,  or  water  containing  finely  divided  particles.  Some  substances, 
such  as  paraffin,  are  homogeneous  and  transparent  when  in  the  fluid  state, 
and  translucent  when  in  the  solid  state.  The  atter  effect  is  apparently 
due  to  granulation  or  crystallization. 

637.  Refraction. — When  light  passes  obliquely  from  one  trans- 
parent medium  to  another  a  part  is  usually  reflected,  while  that 
which  enters  the  second  medium  changes  its  direction  abruptly 
at  the  boundary.     Generally  (but  not  always)  in  passing  from  a 
lighter  medium  to  a  denser  the  light  is 

deflected  toward  the  normal  to  the  bound- 
ary. This  is  called  refraction.  Since  ob- 
jects appear  to  be  in  the  direction  from 
which  the  light  comes,  refraction,  by  chang- 
ing the  course  of  the  light,  causes  an  appar- 
ent displacement  of  the  source.  An  ex- 

i       • •      £          j    •       XT.         i  •  Fl°-  483- 

ample  is  found  in  the  classic  experiment 
of  Kleomedes,  who  showed  that  a  coin  placed  in  the  bottom  of 
a  vessel  so  that  it  is  barely  concealed  by  the  sides  of  the  latter, 
is  apparently  lifted  into  view  when  the  vessel  is  filled  with 
water  (Fig.  483).  The  object  at  A  then  seems  to  be  at  the 
point  A',  and  the  bottom  PQ  of  the  vessel  appears  to  be 
raised  to  P'Q'.  Similarly,  a  meter  rod  dipped  obliquely 
into  water  appears  to  be  bent,  and  the  divisions  seem  to  be 
shortened.  The  latter  effect  is  also  observed  when  the  rod  is 
normal  to  the  surface.  This  change  in  the  apparent  distance 
of  objects  seen  normally  through  a  refractive  medium  is  to  be 
considered  as  an  example  of  refraction,  although  there  is  no  devia- 
tion of  the  light.  It  will  be  shown  in  §667  that  these  effects 
are  the  result  of  differences  of  velocity  of  light  in  the  media 
concerned. 

638.  Intensity  of  Light. — The  brightness  of  light  as  estimated 
by  the  eye  is  not  capable  of  precise  physical  determination.     It 
depends  to  a  large    extent  upon  the  color  of  the  light  and  the 
sensitiveness  of  the  eye.     The  only  consistent  way  in  which 
intensity  of  radiation  may  be  determined  or  expressed  is  in 
terms  of  energy. 

If  radiation  travels  through  a  homogeneous  medium  in  straight 


560  LIGHT 

lines,  and  if  the  medium  is  perfectly  transparent  and  does  not 
itself  emit  radiation,  the  same  total  amount  of  energy  must  flow 
per  second  through  any  spherical  surface  concentric  with  the 
source.  It  follows  that  the  intensity  or  quantity  of  energy 
passing  through  unit  area  per  second,  must  vary  inversely  as 
the  square  of  the  distance  from  the  source  (§259). 

The  above  conclusion  is  based  upon  the  assumption  that  the  radiation 
diverges  uniformly  in  straight  lines  in  all  directions.  It  is  not  true  if  the 
medium  is  of  varying  refractivity,  on  account  of  partial  reflection  and  of 
changing  divergence  of  a  cone  of  light  in  passing  from  one  medium  to 
another.  In  case  a  beam  is  made  parallel  by  a  lens  or  mirror  there  is  no 
change  of  intensity  with  distance  except  that  due  to  absorption  or  to 
imperfect  parallelism. 

639.  Photometry. — The  eye  can  form  no  exact  estimate  of 
degrees  of  intensity,  but  it  can  determine  with  great  accuracy 
whether  two  adjacent  surfaces  are  equally  illuminated  by  lights 
of  the  same  color.  Upon  this  principle  are  based  the  different 
methods  of  Photometry  or  comparison  of 
intensities.  Two  of  the  simplest  and  oldest 
types  of  photometer  are  the  Rumford 
shadow  and  the  Bunsen  grease  spot  photo- 
meters. In  the  use  of  both  it  is  assumed 
that  the  light  from  the  two  sources  com- 
pared contains  the  different  colors  in  the  same  proportions, 
making  comparison  possible. 

In  Rumford's  photometer  shadows  of  a  rod  R  are  cast  on  a 
white  screen  by  the  sources  Sf  and  S2  (Fig.  484) ,  one  of  which 
is  a  standard  comparison  source.  By  adjusting  the  positions 
and  distances  of  $j  and  S2  the  shadows  may  be  made  to  touch 
and  to  be  of  equal  intensity.  When  this  is  the  case,  it  is  evident 
that  the  intensity  of  light  from  each  source  is  the  same  at  the 
screen,  since  each  shadow  is  illuminated  solely  by  the  source 
which  casts  the  other  shadow.  If  this  intensity  is  7,  and  if 
the  intensities  of  th?  sources  at  unit  distance  are  respectively  7, 
and  72, 


)T  the  respective  intensities  of  the  two  sources  are  directly  as  the 


FIQ.  484. 


GENERAL  PROPERTIES  561 

squares  of  their  distances  from  the  screen  when  the  latter  is 
equally  illuminated  by  both.  This  relation  holds  likewise  in 
the  use  of  the  other  forms  of  photometer  described  below. 

The  Bunsen  photometer  consists  essentially  of  a  grease  spot  on 
a  screen  of  white  paper.  Such  a  spot  is  more  translucent  than 
the  clean  paper,  and  for  this  reason  appears  darker  by  reflected 
light  (since  there  is  less  light  reflected  from  the  spot).  If  such 
a  screen  is  placed  between  sources  which  equally  illuminate  it 
with  light  of  the  same  quality  (same  proportions  of  different 
colors)  the  grease  spot  will  disappear.  The  loss  in  light  reflected 
from  the  spot  on  one  side  will  then  be  compensated  by  the 
increased  amount  transmitted  from  the  other  side. 

The  Joly  diffusion  photometer  consists  of  two  rectangular  blocks  of 
paraffin  separated  by  a  piece  of  tin  foil.  Paraffin  is  a  translucent  sub- 
stance which  appears  to  scatter  light  throughout  its  entire  mass.  If  this 
photometer  is  placed  between  two  sources  of  light  with  the  tin  foil  at  right 
angles  to  the  line  joining  them  each  block  will  be  illuminated  by  one 
source  alone.  If  the  intensity  of  illumination  is  the  same  on  both  sides 
the  boundary  line  between  the  two  blocks  will  disappear;  if  it  is  not  the 
same,  the  boundary  is  clearly  seen,  the  block  receiving  the  smaller  amount 
of  light  appearing  darker  than  the  other  throughout  its  entire  mass. 

640.  Lambert's  Law.     A  flat  flame  or  an  incandescent  sheet  of  metal 
appears  to  be  equally  bright  whether  viewed  normally  or  obliquely  to  its 
surface.     The  intensity  or  the  energy  falling  per  second  on  unit  area  of 
the  total  surface  BC  (Fig.  485)  is  equal  to  the  total 
energy  emitted  per  second  from  A  B  at  the  angle 
a  with  the  normal  to  the  surface,  divided  by  BC, 
or  if  Ea   is  the  emisslvity  of  AB  per  unit  area  in 
that  direction,  E=Ea(AB/BC).     The  normal  em  s- 
sivity   is    En  and   observation   shows   that  E***En>  , 
Therefore  A  B 

En=E=Ea(AB/BC)  OT  Ea  «=#»cos  a  Fro-  485. 

Thjs  is   known   as  Lambert's  law.     In  accordance 

with  this  principle,  an  incandescent  sphere  when  viewed  from  a  distance 
appears  to  be  a  uniformly  illuminated  disk. 

The  law  does  not  apply  to  a  surface  bounded  by  an  absorbing  atmosphere, 
which  will  of  course  exercise  greater  total  absorption  in  an  oblique  tfcan  in 
a  normal  direction.  The  sun,  for  example,  which  is  surrounded  by  an 
absorbing  atmosphere  of  gases,  appears  (as  clear  y  shown  in  photographs) 
to  be  darker  at  the  edges  than  at  the  center. 

In  the  same  way  it  may  be  shown  that  if  7ft  is  the  intens'ty  of  light  falling 
normally  on  a  screen,  the  illumination,  when  the  light  is  incident  at  the 
angle  i  is 

/i-J.cosi 

36 


562  LIGHT 

VELOCITY  OF  LIGHT. 

641.  Velocity  of  light. — The  sensation  of  light  is  produced  by  a 
disturbance  originating  in  distant  bodies,  and  it  may  naturally 
be  assumed  that  this  disturbance  travels  with  a  finite  velocity. 
Galileo,  about  1600,  appears  to  have  been  the  first  to  attempt  to 
measure  this  velocity.  His  method  was  substantially  the  same 
as  that  ordinarily  used  to  determine  the  velocity  of  sound  in 
the  atmosphere. 

Two  observers  stationed  at  some  distance  from  each  other  endeavored  to 
note  the  instants  at  which  Sashes  of  light  from  one  station  were  observed  at 
the  other.  The  failure  of  such  attempts  made  it  clear  that  the  velocity  of 

light  is  so  great  that  the  time  required  to 
pass  over  ordinary  distances  is  too  small 
to  be  measured  except  by  methods  much 
more  refined  than  those  at  that  time  avail- 
able. It  was  natural,  therefore,  that  the 
first  results  should  have  been  obtained  by 
astronomical  methods,  in  which  the  distan- 
ces employed  are  those  between  heavenly 
bodies. 

In  1675  Rtfmer,  a  Danish  astrono- 
mer, observed  that  the  eclipses  of 
Jupiter's  satellites  by  that  planet  re- 
cur at  regularly  increasing  or  decreasing  intervals,  according  to 
the  earth's  position  with  respect  to  Jupiter.  If  the  first  obser- 
vations are  made  when  Jupiter  and  the  earth  are  on  the  same 
side  of  the  sun  and  in  line  with  it,  the  interval  between  the 
first  and  the  second  eclipse  of  one  satellite  is  about  1  day  18.5 
hours,  but  as  the  earth  proceeds  in  its  orbit  the  interval  be- 
tween eclipses  slowly  increases,  so  that  when  the  earth  is  on 
the  opposite  side  of  the  sun  from  Jupiter,  the  eclipse  occurs 
about  16  minutes  later  than  the  time  calculated  from  the  first 
observed  interval. 

Homer  explained  this  as  being  due  to  the  finite  velocity  of 
light.  The  distance  between  the  earth  and  Jupiter  having  in 
the  interval  increased  by  the  diameter  of  the  earth's  orbit,  the 
last  installment  of  light  that  comes  from  the  satellite  before 
eclipse  has  this  additional  distance  to  travel  and  in  consequence 
reaches  the  earth  later  by  16m.  41.6s  (according  to  modern 


VELOCITY  OF  LIGHT 


563 


t 


• 


observations) .  This  and  the  best  determinations  of  the  diameter 
of  the  earth's  orbit  give  298,300  kilometers  per  second  as  the 
velocity  of  light. 

642.  Bradley's  Method. — Homer's  explanation  was  discredited 
until  long  after  his  death,  when  an  entirely  different  astro- 
nomical method  confirmed  his  views.  In  1727  Bradley,  the 
astronomer  royal  of  England,  discovered  an  apparent  negative 
parallax  of  the  fixed  stars;  that  is,  an  apparent  displacement 
not  opposite  to  the  direction  in  which  the  earth  was  moving  in 
its  orbit,  but  in  the  same  direction.  Bradley  was  for  a  time 
.greatly  perplexed  by  this  phenomenon,  but  the  chance  obser- 
vation of  the  direction  of  a  wind  vane  on 
a  boat  sailing  on  the  Thames,  this  direction 
not  being  that  of  the  wind,  but  of  the  re- 
sultant of  that  of  the  actual  wind  and  that 
of  the  virtual  wind  due  to  the  motion  of  the 
boat,  suggested  to  him  that  the  apparent 
motion  of  the  light  coming  from  the  stars 
might  be  the  resultant  of  the  actual  motion 
of  the  light  and  its  relative  motion  with  re- 
spect to  the  moving  earth.  If  a  stone  S  (Fig.  5,  T  Tj 
487)  is  dropped  into  a  vertical  tube  T  which 
is  at  the  same  time  moving  parallel  to  itself 
in  a  direction  at  right  angles  to  the  path  of  the  stone,  the 
latter  will  have  a  horizontal  component  of  relative  motion 
with  respect  to  the  tube  and  will  strike  its  side.  When  the 
tube  reaches  TI,  the  stone  reaches  Sj  and  the  relative  path 
of  <S>  with  respect  to  T  is  the  dotted  line  PSlt  Similarly  a 
beam  of  light  which  actually  moves  with  a  finite  velocity 
parallel  to  the  axis  of  a  telescope  tube  will  strike  the  side  of 
the  latter  on  account  of  its  displacement  due  to  the  motion  of 
the  earth.  If  the  apparent  angular  displacement  is  a,  it  is 
evident  that  tan  a**u/V,  where  u  is  the  component  velocity 
of  the  earth  at  right  angles  to  the  line  of  sight  and  V  the  velocity 
of  light.  Bradley  gave  the  name  aberration  to  this  apparent 
angular  displacement  of  the  light  from  the  stars.  The  best 
determinations  of  a,  the  aberration  constant,  is  20.445",  which, 
combined  with  the  known  velocity  of  the  earth  in  its  orbit, 
gives  a  value  for  V  of  299,920  kilometers  per  second 


FIG.  487. 


564 


LIGHT 


643.  Fizeau's  Method. — The  first  to  make  a  direct  determination  of  the 
velocity  of  light  was  Fizeau,  who  in  1849  found  the  time  required  for  light  to 
pass  between  Suresnes  and  Montmartre,  near  Paris,  a  distance  of  8633  meters. 
His  method  was  as  follows:  Light  from  a  source  S  (Fig.  488)  is  reflected 
from  a  piece  of  plate  glass  m,  focused  by  a  lens  L  on  the  circumference  F 
of  a  toothed  wheel  W,  and,  after  passing  between  the  teeth  of  the  wheel,  is 
made  parallel  by  a  second  lens  Lr  From  this  point  the  beam  travels  to  the 
distant  lens  L2,  which  focuses  it  on  a  mirror  M .  From  this  point  the  beam 
retraces  its  path  to  the  source;  but  a  portion  of  it  will  pass  through  the  plate 
glass  m  to  the  eye  E,  by  which  it  may  be  observed.  If  the  toothed  wheel  is 
rapidly  rotated  a  detached  train  of  light  waves  will  pass  through  as  an 


Fio.  488. 

opening  passes  F,  travel  to  M,  and  return.  If  in  the  meantime  a  tooth  has 
moved  into  the  position  F  the  light  will  be  eclipsed;  at  twice  the  speed 
required  for  the  first  eclipse  the  light  will  again  reach  F  when  an  opening 
is  at  the  point,  and  will  pass  to  the  eyepiece!  At  three  times  the  original 
speed  of  the  wheel  the  second  eclipse  will  occur,  and  so  on.  At  speeds 
permitting  transmission  of  the  light  the  waves  will  pass  and  return  through 
the  successive  openings  in  intermittent  groups,  but  the  light  will  appear 
continuous  to  the  eye  because  of  the  persistence  of  vision.  From  the 
distance  between  the  wheel  and  the  distant  mirror  and  the  rate  of  revolu- 
tion of  the  wheel  the  velocity  of  light  can  be  calculated. 

The  value  of  V  found  by  Fizeau  was  313,300  km. /sec.  Cornu,  us'ng  the 
same  method,  obtained  a  mean  result  of  299,950  km. /sec.  from  several 
series  of  experiments. 

644.  Method  of  Foucault,  Michelson,  Newcomb. — In  1862 
Foucault  determined  V  by  means  of  the  displacement  of  a  beam 
of  light  reflected  from  a  revolving  mirror.  The  method  was 
improved  by  Michelson,  who  made  a  series  of  observations  in 
1879  at  the  United  States  Naval  Academy,  and  another  in  1882 


VELOCITY  OF  LIGHT 


565 


in  Cleveland.     Michelson's  arrangement  is  indicated  in  Fig.  489. 

Light  from  a  narrow  slit  S  falls  on  the  mirror  m  and  is  reflected 

to  a  lens  L,  tfhich  throws  it  in  a  parallel  beam  to  the  plane  mirror 

M  .     The  beam  retraces  its  path,  and  if  the  mirror  m  is  at  rest  is 

brought  to  a  focus  at  S.     If,  however,  m  has  rotated  through 

the  angle    a  while  the  light  is  passing  from 

m  to  M  and  back,  the  reflected  pencil  will 

be  rotated  through  the  angle  2  a  and  will  form 

an  image   of  the  source  at  S^     If  the  dis- 

tance between  S  and  iSj  =  d,  that  between  S 

and  m  =  r,  that  between  m  and  M  —  L,  if  n 

be  the  number  of  revolutions  of  m  per  second, 

and  T  the  time  required  for  light  to  pass  from 

m  to  M  and  back, 


m 


Fro.  489. 


2L 

"mt 


Foucault  used  a  short-focus  lens  between  S  and  m  instead  of  a  long- 
focus  lens  between  m  and  M,  as  in  Michelson's  arrangement;  consequently 
L  was  a  short  distance,  not  exceeding  20  meters,  and  the  displacement  d 
was  only  0.7  mm.,  even  when  the  mirror  revolved  800  times  per  second. 
The  result  obtained  by  Foucault  was  298,000  kilometers  per  second.  In 
Michelson's  experiments  a  long-focus  lens  enabled  him  to  make  r  large  and 
at  the  same  time  to  throw  a  parallel  or  nearly  parallel  beam  on  M,  so  that 
the  distance  L  could  be  increased  indefinitely  without  any  considerable  loss 
of  light.  With  a  value  of  L  =  625  meters,  r  =  9  m.,  and  a  speed  of  257 
revolutions  per  second,  the  displacement  d  was  133  mm.  The  result  of 
Michelson's  latest  experiments  in  1882  was  7  =  299,850  km./sec. 

Newcomb,  in  1882,  made  some  further  improvements  in  Foucault's 
method.  The  distance  L  was  3,721  meters,  between  the  Washington  monu- 
ment and  Fort  Myer,  in  Virginia.  The  value  of  V  obtained  by  him  was 
299,860  km./sec.  The  final  results  of  Michelson  and  of  Newcomb  are  prob- 
ably not  in  error  by  more  than  30  km./sec. 

645.  The  Velocity  of  Light  in  Different  Media,  such  as  water 
and  carbon  bisulphide,  was  determined  by  Foucault  and  Fizeau, 
and  also  by  Michelson;  the  method  of  Foucault  being  used  in 
each  case.  A  long  tube  filled  with  the  liquid  was  placed  between 


566  LIGHT 

the  mirrors  m  and  M.  Michelson  found  the  velocity  in  air  to 
be  1.33  times  greater  than  that  in  water,  and  1.76  times  greater 
than  that  in  carbon  bisulphide.  This  has  an  important  bearing 
upon  the  choice  between  the  emission  and  the  undulatory  theo- 
ries of  light  (§§647,  667). 

The  velocity  of  light  from  all  sources  seems  to  be  the  same, 
not  being  appreciably  affected  by  their  intensity.  Romer  and 
Bradley  used  sunlight  or  starlight,  Fizeau  and  Cornu  calcium 
light,  Foucault,  Michelson,  and  Newcomb  sunlight,  Young  and 
Forbes  electric  light.  In  space  lights  of  all  colors  travel  with 
the  same  velocity.  This  is  shown  by  the  eclipse  of  a  white  star 
by  the  moon;  the  star  would  appear  red  just  before  eclipse  and 
blue  just  after  if  blue  light  travels  faster  then  red;  but  no  change 
of  color  is  observed.  It  is  also  shown  by  the  fact  that  in  Michel- 
son's  experiment  the  light  was  not  drawn  out  in  a  spectrum. 
Photographs  of  the  spectrum  of  the  variable  star  Algol,  the  light 
from  which  has  a  period  of  variation  of  about  69  hours,  show  that 
the  intensities  of  the  extreme  violet  and  extreme  red  rise  and 
fall  simultaneously,  proving  that  there  is  no  relative  retardation 
between  them.  In  some  material  media  the  velocity  of  light  of 
different  colors  differs  considerably.  Michelson  found  the 
velocity  of  blue  light  in  carbon  bisulphide  to  be  1.4  per  cent, 
less  than  that  of  red.  In  gases  this  difference  is  inappreciable. 

Light  reaches  the  earth  from  the  moon  in  about  one  second  and  from 
the  sun  in  about  8.25  minutes.  A  small  parallax  has  been  found  in  the 
case  of  some  o"  the  nearer  stars,  which  enables  rough  estimates  of  their 
distances  to  be  made.  Light  from  one  of  the  nearest  stars,  a  Centauri, 
would  require  about  3.75  years  to  reach  the  earth,  and  that  from  Sirius 
about  17  years.  It  seems  quite  possible  that  a  distant  star  may  have  been 
destroyed  by  an  explosion  or  collision  generations  ago,  and  yet  be  visible 
to  us  by  light  emitted  before  its  destruction  and  still  on  its  way  through 
space.  The  changes  frequently  observed  in  variable  stars  must  take  place 
years  before  they  are  evident  to  us. 

THE  NATURE  OF  LIGHT. 

646.  Mode  of  Transmission. — According  to  some  of  the  older 
hypotheses,  such  as  that  of  Descartes,  light  is  the  effect  of  a 
pressure  instantaneously  transmitted  through  a  universal 
medium.  .The  fact  that  the  disturbance  producing  light  has  a 


THE  NATURE  OF  LIGHT  567 

finite  velocity  shows,  however,  that  it  is  due  to  motion,  not  to 
a  static  pressure.  The  radiation  from  such  bodies  as  the  sun 
heats  substances  on  which  it  falls,  and  may  produce  chemical 
changes  or  electrical  effects,  which  shows  that  a  continuous  stream 
of  energy  flows  from  luminous  sources.  According  to  our  ex- 
perience, there  are  only  two  ways  in  which  energy  may  be  trans- 
ferred— by  the  actual  projection  of  material  bodies  through 
space  or  by  the  transmission  of  vibrations  or  pulses  through  a 
stationary  medium,  as  illustrated  by  different  types  of  wave 
motion.  Consequently  there  have  been  two  rival  theories  re- 
garding the  propagation  of  light,  the  emission  theory  and  the 
undulatory  or  wave  theory. 

647.  Emission  Theory. — Sir  Isaac  Newton  believed  that  light 
is  due  to  the  emission  of  luminous  particles  ("corpuscles") 
from  the  source.  He  appears  to  have  adopted  this  hypothesis 
chiefly  because  it  explained  the  rectilinear  propagation  of  light, 
for  which  the  wave  theory  seemed  inadequate.  Newton  showed 
by  prismatic  analysis  that  white  light  is  a  combination  of  many 
different  colors.  He  attributed  difference  of  color  to  difference 
in  size  of  the  corpuscles  exciting  luminosity. 

Newton  observed  that  water  waves  pass  around  obstacles 
without  sensible  disturbance,  casting  no  shadows,  and  that 
sound  shadows  arise  only  under  exceptional  circumstances. 
Reasoning  by  analogy  he  could  not  see  why  light,  if  due  to  wave 
motion,  should  not  travel  around  corners  instead  of  in  straight 
lines.  He  noticed,  however,  that  sound  waves  had  a  greater 
tendency  than  water  waves  to  cast  shadows,  and  if  he  had  care- 
fully observed  the  behavior  of  small  waves,  such  as  ripples  on 
water,  his  objections  to  the  wave  theory  would  probably  have 
been  removed.  While  large  water  waves  pass  around  a  pile 
or  other  comparatively  small  obstacle,  ripples  are  effectually 
stopped,  passing  the  object  on  each  side  without  reuniting;  there 
is  a  well-defined  region  of  no  disturbance,  or  shadow.  Similarly, 
sounds  of  high  pitch,  due  to  very  short  waves,  cast  well-defined 
shadows. 

The  emission  theory  satisfactorily  explains  reflection  if  we  suppose  the 
corpuscles  to  behave  like  elastic  spheres.  If  such  a  sphere  strikes  a  reflecting 
surface  at  an  angle  t  with  the  normal  (Fig.  490)  the  tangential  component  v 
of  its  velocity  will  not  b«  changed.  If  the  magnitude  of  the  reflected  com- 


568  LIGHT 

ponent  u  is  unaltered,  it  follows  that  the  angle  r  of  reflection  is  equal  to  the 
angle  i  of  incidence. 

Refraction  is  also  explained  if  we  assume  that  matter  attracts  these 
particles.  They  will  then  be  subject  to  a  normal  acceleration  as  they 
approach  the  boundary,  while  the  tangential  component  of  velocity  is 
unchanged  (Fig.  491).  If  the  medium  offers  no  resistance  to  the  motion 
of  the  corpuscles  (that  is,  if  it  is  transparent)  it  follows  that  the  increased 
velocity  should  be  maintained  after  entering  the  second  medium,  and  that 


Fio.  490.  Fio.  491. 

the  velocity  of  light  should  be  greater  in  more  refractive  media  than  air 
than  it  is  in  the  latter.  Experiments  show  that  the  opposite  is  true  in  all 
cases  tested  (§645).  This  is  one  grave  objection  to  the  emission  theory. 
Furthermore,  if  matter  attracts  light  corpuscles,  it  would  be  difficult  to 
account  for  the  enormous  expulsive  forces  required  to  project  the  particles 
from  luminous  sources.  We  should  also  expect  the  speed  of  the  particles 
to  vary  with  the  nature  and  activity  of  the  source;  and  yet  the  velocity 
of  light  from  a  candle  appears  to  be  the  same»as  that  from  the  sun. 

648.  Wave  Theory  of  Light. — Huyghens  distinctly  formulated 
this  theory  about  1678.  He  believed  that  space  is  filled  with  a 
rare  medium,  the  ether,  through  which  the  waves  are  propagated 
from  luminous  bodies.  This  theory  accounts  without  any  diffi- 
culty for  the  ordinary  phenomena  of  reflection  and  refraction, 
but  was  not  acceptable  to  Newton  for  the  reason  above  stated. 
For  more  than  a  century  after  Newton's  time  little  progress  was 
made  in  the  subject  of  light,  until,  in  1802,  Thomas  Young 
published  a  paper  "On  the  Theory  of  Light  and  Colors."  In 
this  he  discussed  optical  phenomena  from  the  standpoint  of  the 
wave  theory,  and  first  called  attention  to  the  fact,  overlooked 
by  Huyghens  and  other  advocates  of  the  wave  theory,  that  the 
effect  at  any  point  of  space  through  which  light  waves  are  passing 
is  the  resultant  of  the  effects  of  a  number  of  coincident  individual 
awves.  The  magnitude  of  this  resultant  depends  not  only  on 
the  amplitudes,  but  also  on  the  relative  phases  of  the  component 


THE  NATURE  OF  LIGHT 


569 


waves.  If  two  waves  of  equal  amplitude  and  moving  in  the 
same  direction  are  in  the  same  phase  the  displacement  at  any 
point  is  the  sum  of  the  individual  displacements,  and  the  energy, 
which  is  proportional  to  the  square  of  the  amplitude,  is  four 
times  as  great  as  in  a  single  wave.  If  the  waves  are  opposite 
in  phase,  the  resultant  amplitude  and  energy  at  any  point  are 
zero.  This  effect  Young  called  the  interference  of  light  waves. 

Young  devised  a  simple 
experiment  which  may  be 
regarded  as  a  crucial  test  of 
the  wave  theory.  Light  di- 
verging from  the  slit  S  (Fig. 
492) ,  which  acts  as  a  primary 
source,  passes  through  two 
narrow  slits  Sl  and  $2  very 
close  together,  which  act  as  s 
secondary  sources.  If  a 
screen  be  placed  beyond  these 
slits  a  series  of  colored  and 
dark  bands  parallel  to  the 
slits  will  be  observed  on  it. 
If  one  ef  the  slits  is  covered 
the  bands  disappear.  This 
shows  that  they  are  the  re- 
sultant effect  of  two  superimposed  pencils  of  light  alternately 
reenforcing  and  destructively  interfering  with  each  other.  This 
is  analogous  to  the  interference  of  mercury  ripples  described 
in  §258. 

It  is  easy  to  repeat  Young's  experiment  by  ruling  two  narrow 
slits  very  close  together  on  a  developed  photographic  plate  and 
looking  through  these  slits  at  a  distant  electric  light.  The  expla- 
nation is  as  follows:  Through  the  slits  S,  Slt  and  S2  the  wave 
disturbance  propagates  itself  in  all  directions  beyond  the  respec- 
tive screens  in  semi-cylindrical  waves  having  these  slits  as  axes, 
as  may  be  seen  by  holding  a  white  screen  in  front  of  such  a  narrow 
slit  on  which  light  falls.  It  will  be  seen  that  the  transmitted  light 
diverges  very  considerably  from  the  axis  of  the  pencil,  the  amount 
of  divergence  increasing  as  the  slit  is  narrowed  (§705).  There 
are,  consequently,  when  two  slits  are  used,  two  sets  of  semi- 


Fio.  492. 


570  LIGHT 

cylindrical  light  waves  diverging  from  these  slits  and  crossing 
each  other,  as  shown  in  Fig.  492.  Along  SP0  every  point  of  which 
is  equidistant  from  $,  and  S2,  waves  of  all  lengths  from  the  two 
sources  will  always  meet  in  the  same  phase,  and  there  will  be  a 
maximum  of  white  light  on  the  screen  at  P0.  Along  the  dotted 
line  ending  at  Pj  the  distances  of  any  given  point  from  the  sources 
differ  by  half  a  wave-length;  there  is  destructive  interference 
along  this  line  and  a  minimum  for  the  corresponding  color  at  Pr 
Along  the  line  ending  at  P2  the  difference  between  the  distances 
of  any  given  point  from  the  sources  is  a  whole  wave-length,  so 

that  along  this  line  waves  of  the  same 
length   meet  in  the  same  phase    and 
Pn  there  is  a  maximum  for  the  correspond- 
ing color  at  P2.     At  any  point  Pn  for 
P2  which   /SjPn  —  S2Pn  =  nX    (n   being  any 
whole  number)   there  will  be  a  maxi- 
mum;    where    SlPn-S2Pn  =  ^(2n  +  l)X 
there  will  be  a  minimum. 

Let  P2B  be  equal  to  P2S2.  Then  S1B  =  L  Denote  S,/^  by  a. 
Since  a  is  very  small,  S2B  is  very  nearly  perpendicular  to  /S^Pj 
Hence 

/I  =  asin0  • 

From  this  ^  can  be  deduced.  Measurements  show  that  it  is  very 
small,  being  about  0.000065  cm.  for  red  light  and  0.00004  cm.  for 
violet  light. 

If  AP0  be  denoted  by  D,  since  AP0  and  AP2  are  very  nearly 
equal,  sin  0  =  P0P2/Z>.  Hence  P0P2  =  ZU/a.  Thus  for  a  given 
distance  of  the  screen  the  width  of  a  band  varies  directly  as  the 
wave-length  and  inversely  as  the  distance  between  the  slits. 

649.  Relation  between  Color,  Wave-length,  and  Frequency. — If 
white  light  falls  on  the  slits  the  inner  side  of  each  band  is  violet, 
the  outer  side  red.  This  shows  that  the  wave-length  is  different 
for  light  of  different  colors,  and  that  the  wave-length  of  violet 
light  is  less  than  that  of  red.  The  central  band  is  of  course  white, 
as  all  colors  have  a  maximum  at  this  point,  regardless  of  their 
wave-length.  From  the  relation  nX  —  V  (§246)  where  n  is  the 
frequency  of  vibration,  X  the  wave-length,  and  V  the  velocity  of 
light,  it  is  evident  that  when  V  changes  either  n  or  X  or  both  must 
change.  If  Young's  experiment  be  performed  in  a  medium  such 


THE  NATURE  OF  LIGHT 


571 


as  water,  it  is  found  that  the  width  of  the  bands  in  water  is  to 
their  width  in  air  as  the  velocity  of  light  in  water  is  to  that  in  air. 
Hence  XJX  =  Vl/  V,  and  n  is  constant.  It  is  a  matter  of  common 
experience  that  the  color  of  a  beam  of  light  does  not  change  when 
it  enters  water,  hence  frequency  rather  than  wave  length  deter- 
mines color.  Color  is,  therefore,  analogous  to  pitch  in  sound. 

660.  The  Ether. — To  account  for  the  transmission  of  waves 
through  space  containing  no  ordinary  matter  it  seems  necessary 
to  assume  the  existence  of  a  universal  medium  filling  all  space 
and  even  interpenetrating  matter  itself,  as  shown  by  the  existence 
of  transparent  substances.  That  this  medium  can  react  on 
matter  is  shown  by  the  fact  that  radiant  energy  is  transmitted 
from  ether  to  matter  in  the  case  of  absorption,  and  from  matter 
to  ether  in  the  case  of  emission  of  radiation  by  material  sources. 

In  recent  years  doubt  as  to  the  necessity  for  assuming  the  existence  of 
an  ether  has  been  expressed  by  some  who  believe  that  it  is  sufficient  to 
attribute  the  power  of  transmitting  radiation  to  space  itself.  It  may  be 
doubted  whether  this  is  more  than  a  dispute  about  terms.  We  cannot 
discuss  the  question  here,  but  pending  the  settlement  of  the  controversy 
it  seems  wise  to  continue  the  use  of  the  word  ether  as  at  least  denoting 
the  power  of  space,  vacant  or  occupied  by  matter,  to  transmit  radiation. 

651.  Huyghens'  Principle. — Huyghens  assumed  that  a  wave  is 
propagated  by  every  point  of  the  medium  in  a  wave  front  acting 
as  a  new  center  of  disturbance  as  has  already 
been  explained  and  illustrated  in  the  case  of 
water  waves  (§256  and  Fig.  166/).  There- 
suiting  wave  front  is  the  enveloping  tangent 
plane  to  the  wavelets  starting  from  these  cen- 
ters, as  shown  in  Fig.  494. 

The  points  a,  6,  c,  etc.,  between  A  and  B 
(Fig.  484)  taken  as  close  together  as  we 
please,  act  as  centers  of  disturbance.  Along 
the  tangent  plane  A'B'  the  different  waves 
are  all  in  the  same  phase,  and  each  point  in 
this  new  tangent  plane  becomes  a  new  cen- 
ter of  disturbance,  so  that  the  resultant 
wave  travels  forward  as  rapidly  as  the  disturbance  is  propagated 
from  point  to  point  of  the  medium. 

The  waves  move  forward  without  hindrance,  because  there  is  no  existing 
displacement  to  oppose  them ;  they  do  not  travel  backward,  because  there 


FIG.  494. 


572  LIGHT 

is  a  force  due  to  the  existing  displacement  on  the  side  from  which  the 
waves  come  sufficient  to  nullify  the  backward  component  of  the  displace- 
ment due  to  each  successive  center  of  disturbance.  It  is  like  the  propa- 
gation of  a  shove  through  a  line  of  people,  or  of  elastic  spheres  of  the  same 
mass  and  elasticity;  that  in  front  is  not  braced  to  withstand  the  impulse, 
while  the  reaction  on  the  one  communicating  the  impact  is  expended  in 
overcoming  its  forward  momentum.  ^ 

652.  Origin  and  Properties  of  Light  Waves. — Sources  of  light 
are  usually   bodies   of  high   temperatures.     According  to  the 
mechanical  theory  of  heat,  high  temperature  corresponds  to  a 
violent  agitation  or  vibration  of  the  ultimate  particles  (molecules, 
atoms  or  electrons)    (§159)  of  matter.     We  may  imagine  that 
these  particles  impart  their  motion  to  the  surrounding  ether 
in   much  the  same  way  that  a  tuning  fork  generates  sound 
vibrations  in  air. 

So  far  no  evidence  has  been  presented  to  show  whether  these 
waves  are  longitudinal,  like  those  of  sound,  transverse,  like 
those  in  a  stretched  wire,  or  of  a  more  complex  character,  like 
water  waves.  In  §753  it  will  be  shown  that  the  displacements 
in  these  waves  must  be  transverse  to  the  direction  of  propagation. 

We  may  now,  as  a  working  hypothesis,  assume  that  light  is 
due  to  transverse  periodic  displacements  in  a  universal  medium, 
set  up  by  the  agitation  of  the  ultimate  particles  of  matter,  that 
these  waves  are  of  different  lengths  (periods  of  vibration),  but 
are  all  very  short;  that  different  colors  correspond  to  different 
rates  of  vibration;  and  that  waves  of  all  lengths  travel  with  the 
same  velocity  in  free  space,  but  with  different  velocities  in 
matter.  All  experimental  facts  are  in  harmony  with  these 
assumptions. 

In  the  following  pages  the  word  ray  will  often  be  used  as  a 
matter  of  convenience,  meaning  thereby  merely  a  normal  to  the 
wave  front,  which  indicates  the  direction  in  which  the  wave  is 
moving  at  the  point  considered.  The  definition  applies  only  to 
isotropic  media  (§§163,  761). 

REFLECTION. 

653.  Reflection  from  a  Plane  Surface. — A  wave  diverging  from 
the  source  S  (Fig.  495)  falls  on  a  plane  mirror  MN.     If  the  mirror 
were  absent,  the  wave  would  at  a  given  instant  occupy  the  posi- 


REFLECTION 


573 


tion  AMPNB.  With  the  mirror  in  place,  each  element  of  the 
original  wave  when  it  reaches  the  mirror  becomes  the  center  of  a 
reflected  wavelet,  just  as  it  would  have  contributed  a  wavelet 
from  the  same  point  to  form  the  resultant  wave  MPN  if  the 
mirror  were  absent.  If,  therefore,  a  number  of  circles  tangent 
to  MPN  be  described  about  centers  a,  b,  c,  etc.,  on  MCN  they 
must  touch  both  the  imaginary  wave  MPN  and  the  reflected 
wave  MQN.  The  arcs  MPN  and  MQN  are  evidently  similar 
and  equal  and  have  equal  radii  of  curvature.  If  Sl  is  the  center 
of  curvature  of  the  reflected 
wave  SC^CSi,  the  line  SSl 
is  normal  to  the  mirror,  and 
Si  is  as  far  behind  the  latter 
as  S  is  in  front  of  it.  If  the 
eye  is  at  E,  any  point  reached 
by  the  reflected  wave,  the 
pencil  of  light  entering  the 
pupil  will  be  focused  on  the 
retina.  As  the  vertex  of  this 
cone  is  virtually  at  Slt  the 
image  of  the  source  will  ap- 
pear to  be  at  that  point. 
From  the  diagram  it  is  evident 
that  the  angles  i  and  r  are  equal,  that  is,  the  angles  of  incidence 
and  of  reflection  are  equal. 

654.  Focus. — The  source  or  center  of  curvature  of  a  family  of 

waves,  either  divergent  or  convergent, 
is  called  a  focus — literally  a  hearth  or 
source  of  radiation.  The  point  S 
from  which  the  waves  actually  come 
is  called  a  real  focus;  the  point  Sl 
from  which  they  appear  to  come  is 
called  a  virtual  focus.  The  points 
S  and  Si  are  conjugate  foci.  Since  the 
conjugate  focal  distances  in  the  case 
of  a  plane  mirror  are  equal,  it  is  evi- 
dent that  if  the  mirror  be  displaced 
a  given  distance  parallel  to  itself  the  image  will  be  displaced 
twice  that  distance. 


Fio.  495. 


574 


LIGHT 


655.  Images.— If  A'B'  is  the  image  of  AB  (Fig.  496),  it  may  be 
shown  as  above  that  the  image  of  each  point  is  as  far  behind  the 
mirror  as  the  point  itself  is  in  front,  and  on  the  same  normal;  and 
that,  consequently,  the  image  and  the  object  are  symmetrically 
placed  with  respect  to  the  mirror  and  are  of  the  same  size. 


Fio.  497. 


FIG.  498. 


656.  Multiple  Reflection. — Fig.  497  shows  how  these  images  are  situated 
in  the  case  of  multiple  internal  reflection  from  surfaces  AB  and  CD  parallel 
to  each  other.     The  position  of  these  images  is  readily  determined  by  the 
fact  that  the  image  of  the  first  order  in  each  surface  is  as  far  behind  the 
surface  as  the  source  is  in  front,  and  on  the  same  normal  to  the  surface. 
The  two  images  of  the  second  order  are  fixed  in  the  same  way,  by  considering 
the  images  of  the  first  order  to  be  the  sources,  and  so  on  ad  infinitum. 
It  is  easy  to  see  that  when  the  mirrors  are  inclined  at  an  angle  a  (Fig.  498) 
there  are  multiple  images  of  the  mirrors  as  indicated  by  the  dotted  lines, 
and  that  the  successive  images  are  symmetrically  placed  on  each  side  of 
each  mirror  image  and  located  in  a  circle  about  the  point  of  intersection 
of  the  mirrors. 

657.  -Reflection  from  Curved  Surfaces. — If  a  wave  is  reflected 
from  a  curved  surface  the  curvature  of  the  reflected  wave  is 
changed,  unless  it  exactly  conforms  to  the  mirror  surface  at  inci- 
dence.    Experience  shows  that  only  in  a  few  cases  is  the  re- 
flected wave  spherical  or  approximately  so,  and  only  in  such  cases 
can  a  definite  image  be  formed.     The  ordinary  type  of  curved 
mirror  is  that  with  a  spherical  surface.     The  reflected  waves  are 
approximately  spherical  if  the  diameter  of  the  mirror  is  small 
compared  with  its  radius  of  curvature.     In  order  to  determine 


REFLECTION 


575 


the  position  of  the  center  of  curvature  and  the  conjugate  focal 
relations  for  spherical  mirrors  a  very  simple  mathematical 
relation  is  all  that  is  required. 

658.  Relation  between  Radius  of  Curvature 
and  Sagitta  of  Arc. — Consider  the  arc  A  B, 
with  center  of  curvature  C,  and  radius  r  (Fig. 
499).  The  distance  x  on  the  bisecting  radius 
of  the  arc  included  between  the  arc  and  the 
chord  AB  =  2y  is  called  the  sagitta  of  the 
arc.  To  determine  the  relation  between  r  and 
x  write 


Therefore, 


Fio.  499. 


2r  —  x    r(l  +  cos  «) 

It  is  found  that  if  the  angle  a  is  very  small,  not  more  than  two  or 
three  degrees,  the  mirror  will  give  a  well-defined  image.  If  the 
angular  aperture  2  a  of  the  mirror  is  greater  than  four  or  five 
degrees  spherical  aberration  becomes  noticeable  (§664).  For  all 
mirrors  which  give  satisfactory  images  x  may  be  neglected  in 
comparison  with  r,  or  cos  a  regarded  as  equal  to  unity,  so  that 
within  the  limits  of  errors  of  measurement 


2r 

669.  Concave  Mirror. — The  source  is  at  a  distance  u  from  a 

concave  mirror  MN  (Fig.  500) 
with  center  of  curvature  at  C 
and  radius  r.  The  waves  inci- 
dent on  the  mirror  have  a  ra- 
dius of  curvature  u,  with  a 
sagitta  AB.  Reflection  begins 
at  M  and  N  while  the  vertex  of 
the  wave  has  still  to  travel  the 
distance  BD  before  reflection 
begins  at  D.  When  the  vertex 
reaches  the  mirror  the  edges  of 
the  wave  have  travelled  a  distance  BD  —  AD  —  AB  along  MSl 


and 


If  the  reflected  wave   is  spherical  it  must  have  a 


576  LIGHT 

definite  center  of  curvature  Sl  and  radius  v,  with  sagitta  DE. 
At  the  instant  when  reflection  begins  at  D  the  incident  wave, 
the  mirror,  and  the  reflected  wave  have  a  common  point  of 
tangency  at  D.  If  the  angular  aperture  of  the  mirror  is  so  small 
that  the  cosines  of  the  angles  a,  /?,  and  7-  may  be  considered  as 
equal  to  unity  (these  angles  are  exaggerated  in  the  figure  for  the 
sake  of  clearness)  we  may  consider  the  portions  of  the  wave 
reflected  from  M  and  N  to  move  parallel  to  the  axis  rather  than 
in  the  directions  MSl  and  NS^  Hence 

AD-AB=DE-AD 

AB+DE  =  2AD 

It  is  not  convenient  to  measure  sagittse,  but  by  using  the 
relation  developed  in  §658  the  above  expression  can  be  trans- 
formed into  one  involving  only  the  easily  measured  distances  r, 
the  radius  of  curvature  of  the  mirror,  u,  that  of  the  incident  wave, 
and  v,  that  of  the  reflected  wave.  The  semi-chord  y  has  the 
same  value  for  all  the  arcs  concerned,  so  that  the  common  factor 
y*/2  may  be  cancelled  when  y*/2r  is  substituted  for  AD,  with 
similar  substitutions  for  AB  and  DE.  The  final  result  is 

1+1.2 

u      v      r 

The  justification  for  the  somewhat  inexact  assumptions  made 
in  deriving  this  formula  is  found  in  the  fact  that  it  agrees  with 
experimental  observations  within  the  limits  of  error  of  measure- 
ment. 

A  beam  of  light  is  always  reversible  in  direction,  hence,  if  the 
source  is  at  Slt  the  image  will  be  at  S. 

If  the  source  is  at  a  great  distance  from  the  mirror  the  incident 
wave  is  practically  plane  (parallel  beam),  and  u  is  infinite.  The 
corresponding  value  of  v  is  called  the  principal  focal  distance  f. 
Si  is  then  the  principal  focus.  The  above  equation  then  becomes 

1,12  r 

-+7~r    or    /.g 

Hence  the  principal  focal  point  is  half  way  between  the  mirror 
and  its  center  of  curvature.  The  conjugate  focal  relation  may 
now  be  written: 

1+1*1 

u+v    f 


REFLECTION  577 

If  u  >  r,  v  <  r.     The  image  is  between  C  and  the  mirror. 

If  u=r,  v  =  r.     The  image  is  at  the  center. 

Ifw<r,  v>r.     The  image  is  beyond  C. 

If  u  =/,  v  =  oo  .  The  reflected  light  is  parallel. 

If  u<f,  v  is  a  negative  quantity.  Fig.  501,  which  illustrates 
this  case,  shows  that  the  center  of  curvature  of  the  reflected 
wave  is  behind  the  mirror. 
It  is  a  virtual  focus,  since 
the  waves  do  not  actually 
diverge  from  that  point. 
It  is  clear  that  the  nega-  C 
tive  sign  of  v  indicates 
this  result,  since  the  dis- 
tance DSi  =  v  is  measured 
in  a  direction  opposite  to  that  in  which  light  actually  proceeds 
after  reflection,  so  that  the  reflected  light  cannot  pass  through 
the  point  S^ 

Writers  differ  in  their  conventions  regarding  the  signs  of  con- 
jugate focal  distances  and  radii  of  curvature.  The  most  easily 
remembered  and  applied  as  well  as  the  most  consistent  rule  seems 
to  be  the  following: 

Consider  each  of  the  quantities  u,  v,  r  as  positive  when  it  is  on 
the  same  side  of  the  mirror,  as  in  the  typical  case  of  a  concave 
mirror  forming  a  real  image  of  a  real  object — negative  when  on  the 
opposite  side. 

From  this  rule  it  is  evident  that  a  positive  value  of  v  or  / 
indicates  a  real  focus,  a  negative  value  a  virtual  focus. 

660.  Convex  Mirror. — Proceeding  as  in  the  previous  case,  if 
FG  is  the  sagitta  of  the  reflected  wave  and  v  its  radius  (Fig.  502), 

DE+EF=FG-EF 

DE-FG=-2EF 

!__!_  _2_  _1 

u     v"       r"      / 

where,  provisionally,  v,  r,  and  /,  may  be  considered  as  mere 
magnitudes  affected  with  the  negative  signs  in  the  formula. 

Comparing  this  expression  with  that  deduced  for  a  concave 
mirror,  we  see  that  it  will  become  identically  the  same  if  we 
agree  to  consider  the  radius  of  curvature  and  the  principal  focal 

37 


578 


LIGHT 


distance  of  a  convex  mirror  as  negative  in  accordance  with  the 
rule  given  above;  v  is  also  negative  since  the  reflected  light  is 
divergent. 

The  general  formula  applicable  to  all  mirrors  is,  therefore, 


W      V      / 

u  being  usually  positive,  and  /  to  be  taken  as  positive  for  a  con- 
cave, negative  for  a  convex  mirror.  When  /=  oo  we  have  the 
case  of  a  plane  mirror. 


FIG.  502. 

If  we  make  /  negative  in  the  expression  for  v  given  in  §659, 
we  see  that  in  the  case  of  a  convex  mirror  v  is  always  less  than  / 
and  negative.  If,  however,  the  light  incident  on  the  mirror  is 
convergent  to  a  point  at  a  distance  —  u  behind  the  mirror,  v  may 
become  positive;  so  that  a  con  vex  mirror  may  give  a  real  image 

of  a  virtual  source. 

661.  Geometrical  Method. — 
The  same  results  may  be  ob- 
tained by  applying  the  law  of 
plane  reflection  to  "rays"  with- 
out any  hypothesis  as  to  the 
nature  of  light.  The  ray  *SD 
(Fig.  503)  will,  as  it  is  incident 

normally  at  D,  be  reflected  back  on  itself.  The  ray  SP  will  be 
reflected  at  P,  so  that  the  angles  i  and  r  are  equal.  The  in- 
tersection of  these  two  reflected  rays  will  fix  the  position  of  the 
image  Si.  From  a  well-known  geometrical  relation  we  have 


FIG.  603. 


SC      SD-CD    u-r 


u 


sn 


sn 


sn       sn 


Therefore, 


REFLECTION 
CSi     CD-SiDr- 


579 


sn  r 


sn  r 


sn  r    sn 


u—r_u 
r—v     v 


From  which 

Dividing  through  by  uvr, 


u 


The  assumptions  made  in  this  case  are  that  SD  =  u  and  SiD  =  v, 
which  is  a  sufficient  approximation  to  the  truth  when  the  angles 
a,  0,  and  d  are  small. 

The  formula  for  the  conjugate  focal  relations  of  a  con  vex  mirror 
may  be  derived  in  the  same  way. 

In  some  cases  it  is  more  convenient  to  use  the  geometrical  or 
ray  method  than  that  of  waves;  but  it  must  always  be  remem- 
bered that  these  "rays"  merely  represent  normals  to  the  wave 
front. 

662.  Images  Formed  by  Spherical  Mirrors. — If  any  two  radii 
be  drawn  from  any  point  of  a  source,  the  point  of  their  inter- 
section after  reflection  will  fix  the  position  of  the  corresponding 


Fia.  604. 

point  of  the  image.  Any  pair  of  radii  will  do,  but  for  convenience 
two  of  the  following  are  usually  chosen,  because  their  course 
after  reflection  is  easily  determined:  The  radius  parallel  to  the 
axis,  which  after  reflection  passes  through  the  principal  focus; 
that  which  passes  through  the  principal  focus,  which  becomes 
parallel  to  the  axis  after  reflection;  that  which  is  incident  at  the 
intersection  of  the  mirror  with  its  axis. 


580 


LIGHT 


The  construction  of  the  images  formed  by  a  concave  and  by 
a  convex  mirror  is  illustrated  by  Figs.  504,  505,  506,  where 


the  points  A'  and  B'  are  located  by  using  the  pair  of  rays  last 
mentioned  above.  In  the  first,  the  image  is  real  and  inverted; 
in  the  second  and  third  the  images  are  virtual  and  erect. 


Fio.  506. 

663.  Magnification. — Since  the  angle  subtended  by  the  object 
at  the  mirror  is  i,  while  that  subtended  by  the  image  is  the  equal 
angle  r,  it  is  evident  that  the  relative  sizes  of  the  object  and 
image  are  to  each  other  as  their  respective  distances  from  the 
mirror. 

o/i  =  u/v 

The  real  image  formed  by  a  concave  mirror  may  be  of  the  same 
size  as  the  object,  or  larger,  or  smaller;  the  virtual  image  is  always 
larger,  since  v  >  u.  The  virtual  image  formed  by  a  convex  mirror 
is  always  smaller  than  the  object. 

664.  Spherical  Aberration  and  Caustic  Curves. — If  a  converging 
wave  is  truly  spherical  there  is  a  perfect  focus  at  its  center  of 
curvature.     As  a  matter  of  fact,  the  waves  reflected  from  a 
spherical  mirror  are  not  perfectly  spherical,  except  in  the  special 
case  where  the  source  is  at  the  center  of  curvature  of  the  mirror. 
The  normals   drawn  from  any   point?  of  the  reflected   wave 


REFLECTION 


581 


Fio.  507. 


are  tangent  to  the  curve  HF,  which  is  called  a  caustic  curve. 
The  cusp  F  of  this  curve  corresponds  to  the  focal  point  of  a 
mirror  of  small  aperture.  The  light  reflected  from  the  sides  of 
a  cup  containing  coffee  or  milk  plainly  shows  this  caustic  curve 
on  the  surface  of  the  liquid. 

The  deviation  from  a  spher- 
ical shape  of  waves  reflected  from 
a  mirror  of  large  aperture  is 
called  spherical  aberration. 

If  light  is  obliquely  incident 
on  a  mirror,  the  reflected  waves 
are  not  spherical,  even  when 
the  aperture  is  small,  but  have 
different  radii  of  curvature  in 
planes  at  right  angles.  As  a 
result,  a  point  image  of  a  point 
source  cannot  be  obtained,  but 
there  are  two  elongated  images  at  right  angles  to  each  other  and 
in  different  positions,  which  are  called  focal  lines. 

The  origin  of  the  focal  lines  is  clearly  seen  if  we  consider  tne  mirror  M N 
(Fig.  508)  to  be  part  of  a  larger  mirror  MNO,  on  the  axis  SB  of  which  the 
source  S  lies.  Constructing  the  reflected  rays  incident  at  different  points 

of  this  mirror,  it  is  clear  that,  while  the 
focal  cusp  of  the  entire  mirror  is  at  Sv 
all  the  rays  coming  from  M  N  intersect 
approximately  at    the  point  Fv    The 
diagram  gives  a  cross-section  of  the  in- 
cident  and  reflected  rays.     If  this  dia- 
gram be  rotated  about  the  axis  SB  by 
an  amount  equal  to  the  diameter  of  the 
/        mirror  M  N  the  point  F^  will  describe  the 
0/          arc  of  a  circ'e  with  its  center  on  the 
Fio.  508.  line  SB.     This  is  the  primary  focal  line, 

which  will  appear  on  a  screen  placed  at 

Fl  as  a  narrow  curved  strip.  After  passing  Fl  all  the  rays  reflected  from 
M  N  will  intersect  the  axis  SB  at  various  points  between  Sl  and  F3  (since 
all  the  planes  of  incidence  contain  SB).  A  screen  placed  at  this  point  will 
show  a  narrow  elongated  patch  of  light,  <S1F2,  the  secondary  focal  line.  If 
the  screen  is  at  right  angles  to  the  reflected  pencil  the  patch  of  light  will  be 
approximately  a  lemniscate  or  figure  8. 

666.  Cylindrical  Mirror. — A  parallel  beam  incident  on  such  a  surface  is 
brought  to  a  real  or  virtual  line  focus.  The  image  of  a  point  source  is 


582 


LIGHT 


likewise  a  line.  Such  mirrors  and  the  reflected  pencil  are  said  to  be  astig- 
matic. (A  pencil  symmetrical  about  an  axis,  that  is,  having  a  point  vertex, 
and  thus  giving  a  point  image  of  a  point,  is  said  to  be  homocentric.)  In 
the  case  of  a  concave  cylindrical  mirror,  if  the  point  source  lies  outside 

the  principal  focus,  there  will  be  a 
real  image  AB  and  a  virtual  image 
A'B'  in  planes  at  right  angles  to 
each  other,  as  illustrated  in  Fig.  509. 
666.  Paraboloidal,  Ellipsoidal,  and 
Hyperboloidal  Mirrors. — The  light 
from  a  point  source  at  one  focus  of 
an  ellipsoidal  reflector  will  be  brought 
without  aberration  to  the  other 
focus,  a  real  image  being  formed. 
Light  from  a  source  at  one  focus 
of  a  hyperboloidal  mirror  will  have 
a  virtual  focus  at  the  conjugate 
Fio.  509.  focus  of  the  mirror.  If  the  source 

is   at   the  focus  of  a  paraboloidal 

mirror,  the  light  will  be  reflected  in  a  parallel  beam;  and  parallel  light  will 
be  brought  without  aberration  to  a  real  focus  by  such  a  mirror. 

REFRACTION  AND  DISPERSION 

667.  The  ancients  were  acquainted  with  the  fact  that  a  beam  of 
light  is  more  or  less  deviated  in  passing  from  air  to  water.     The 


V 

B' 


N 


Fio.  610. 


Law  of  Refraction  was  first  discovered  in 
1621  by  Willebrod  Snell.  He  found  by  ex- 
periment  that  the  ratio  of  the  sines  of  the 
angles  of  incidence  and  of  refraction  is  con- 
stant at  the  boundary  between  two  media. 
The  ratio  sin  I'/sin  r  is  called  n,  the  index 
of  refraction.  The  angle  of  incidence  is 
usually  measured  in  air. 

It  was  shown  by  Huyghens  that  refraction  is  very  simply 
explained  by  assuming  a  change  of  velocity  in  passing  from  one 
medium  to  another.  Direct  measurements  by  Foucault,  Fizeau, 
and  Michelson  show  that  light  travels  with  different  velocities 
in  air,  water,  and  carbon  bisulphide  (§645). 

Consider  a  plane  wave  AC  incident  obliquely  on  the  smooth 
plane  surface  of  separation  between  air  and  another  trans- 
parent medium  (Fig.  511),  the  velocity  in  air  being  Vl  and  that 
in  the  second  medium  V2.  A  spherical  wave  will  diverge  from 
the  point  A  into  the  second  medium  when  the  disturbance 


REFRACTION  AND  DISPERSION 


583 


reaches  that  point,  and  later  other  spherical  waves  successively 
diverge  from  B'  and  C".  While  the  wave  travels  in  the  first 
medium  a  distance  CC'  =  Vlt  the  wave  from  A  will  travel  the 
distance  AA* ' —  V \t  in  the 
second  medium.  The  dis- 
turbance from  B  will  in  the  \£ 
same  time  travel  a  distance 


if   B  is  half  way  between 

A  and  C.     Since   B'B"  = 

JAA',  a  tangent  plane  can 

be  drawn  from  C'  to  the  two 

circles   with  centers  at  A 

and  B'.     It  is  easily  shown 

by   this   method  that  the 

waves  from  all  points  in  the  original  wave  front  will  be  tangent 

to  the  same  plane,  the  new  wave  front  in  the  second  medium. 

Further, 


Therefore, 


sn 


sin  i    V. 
-  —  —  ^—  n 
smr    7 


The  physical  significance  of  the  constancy  of  the  sine  ratio 
discovered  by  Snell  thus  becomes  apparent.  The  student  should 
always  think  of  the  index  of  refraction  as  being  the  ratio  of  the 
velocities  of  light  in  the  two  media,  rather  than  as  the  ratio  of 
the  sines  of  two  angles.  The  latter  mode  of  statement  conveys 
no  clear  physical  idea,  and,  moreover, 
seems  to  break  down  in  the  case  of 
normal  incidence. 

668.  Medium  with  Parallel  Surfaces. 
— An  incident  pencil  will  be  deflected 
in  one  direction  on  entering  the  second 
medium  of  thickness  t  and  an  equal 
amount  on  ree'ntering  the  first  medium, 
as  shown  in  Fig.  512.  The  course  of 
the  pencil  will  then  be  parallel  to  its  original  direction,  but 
there  will  be  a  lateral  displacement  AB. 


FIQ.  512. 


584 


LIGHT 


669.  Image  due  to  Refraction  at  Plane  Surface. — When  an 
object  is  viewed  normally  to  the  boundary  (Fig.  513)  there  is  no 
lateral  displacement,  but  only  an  apparent  change  in  distance. 
Waves  from  an  object  S  at  a  distance  d  below  the  Surface  of  the 

medium  travel  with  a  velocity  V2  to 
the  point  B,  where  the  vertex  of  the 
wave  enters  air,  in  which  the  velocity  is 
7r  The  disturbance  then  travels  a 
distance  BE'  in  air  while  another  por- 
tion of  the  wave  still  within  the  second 
medium  travels  the  distance  A  A'  =  V2t. 
The  center  of  curvature  of  the  emer- 
gent wave  is  at  Sl}  a  distance  dl  below  the  surface.  There  is  a 
virtual  image  of  the  source  at  this  point.  If  the  cone  has  only 
a  small  divergence,  AA'  =  BC,  the  sagitta  of  the  wave  in  the 
refracting  medium,  BB'  is  that  of  the  wave  in  air,  and  d  =  AS 
and  dl  —  ASltheiT  respective  radii  of  curvature;  hence,  from  the 
relation  previously  used  (§658). 


Therefore, 


d     V, 
= 


The  angle  of  the  cone  of  light  entering  the  eye  is  limited  by  the 
size  of  the  pupil,  and  is,  therefore,  very  small,  so  that  the  use  of 
the  above  method  is  justified.  The 
apparent  depth  of  the  object  below 
the  surface  is  dl  =  d/n.  There  is  an 
apparent  displacement  toward  the 
observer  amounting  to  (d  —  d^  = 
(n—l)d/n.  It  is  thus  made  clear  why 
the  depth  of  a  pond  appears  to  be  less 
than  it  actually  is,  and  why  objects 
immersed  in  water  appear  to  be 
shortened.  Since  the  index  of  refrac-  FlQ  514 

tion  is  about   1.33,  a  pond  six  feet 
deep  seems  to  be  only  about  four  and  a  half  feet  in  depth. 

If  the  cone  is  wide  there  is  considerable  aberration,  as  shown 


REFRACTION  AND  DISPERSION 


585 


in  Fig.  514.  This  is  not  apparent  to  the  eye,  which  limits  the 
aperture  of  the  effective  pencil,  except  through  a  slight  lateral 
displacement  (the  image  being  at  Q  if  the  eye  is  at  E) . 

The  index  of  refraction  of  plane  parallel 
plates  may  be  obtained  from  the  relation 
deduced  above.  A  microscope  is  focused 
on  a  small  object  on  a  table,  such  as  a 
pencil  mark  0  (Fig.  515).  When  the 
plate  is  placed  over  the  mark  it  will  be 
necessary  to  raise  the  microscope  a  dis- 
tance d  to  bring  the  virtual  image  o  into 
focus.  The  apparent  depth  of  the  object 
below  the  surface  is  t'  =  t/n,  and  d  =  t-t'  =  t-t/n.  Hence 


0 

Fia.  515. 


Fio.  516 


t-d 

670.  Prism. — If  light  waves  pass  through  a  transparent  medium 
bounded  by  plane  surfaces  which  are  not  parallel,  the  deviation 
of  the  incident  pencil  on  entering  the  first  surface  is  not  exactly 

compensated  on  emerging  from  the 
second  surface.  If  the  source  is  at  S 
;.  516)  the  image,  or  center  of  cur- 
vature of  the  wave  within  the  prism, 
is  at  Sl  and  that  of  the  emergent  wave 
is  at  S2.  To  determine  the  deviation 
of  the  pencil  and  the  positions  of  the  foci  St  and  S2  it  is  convenient 
to  follow  the  course  of  given  wave  normal  or  "  ray."  The  inter- 
section of  pairs  of  such  rays  will'  fix  the  position  of  the  desired 
foci  or  centers  of  curvature  of  the  waves. 

671.  Deviation — Minimum  Devia- 
tion.—The  total  deviation  of  a  given 
ray  is  D  =  Dl  +  D2  (Fig.  517). 


D  =i1+;2-( 

But  r-fr  =  A   since 


FIQ.  517. 


Therefore,  D  =it  +i2-  A 
It  is  easily  shown,  experimentally  or  mathematically,  that  D 


586  LIGHT 

has  a  minimum  value  when  t1=t2;  in  which  case  the  incident  and 
emergent  ray  are  symmetrical  with  respect  to  the  refracting  angle 
of  the  prism.  In  this  case 

.      . D+A 


Therefore, 

sin  i 

sin  r         sin  J  A 

This  relation  is  commonly  used  for  determining  the  index  of 
refraction  of  substances  ;'n  the  prismatic  form.  The  angles  of 
the  prism  and  of  minimum  deviation  are  measured  with  a  spectro- 
meter (§720).  As  the  index  is  not  the  same  for  different  colors, 
it  is  evident  that  the  prism  can  be  set  at  the  angle  of  minimum 
deviation  for  only  one  color  at  a  time. 

*  672.  Dispersion  of  Color, — The  index  of  refraction  of  any  given 
substance  varies  with  the  color,  or  wave  length;  consequently  the 
deviation  caused  by  a  prism  will  not  be  the  same  for  all  colors. 


Fio.  518. 

Consider  a  narrow  source  S  such  as  an  illuminated  slit  parallel 
to  the  edge  of  the  prism  (Fig.  518).  If  the  source  emits  red  light 
alone,  a  virtual  red  image  of  the  slit  is  observed  at  R.  If  green 
and  violet  light  are  also  emitted,  a  green  and  a  violet  image  are 
seen  at  G  and  V.  Real  images  of  these  colors  may  be  formed  at 
R' ,  G',  and  V  by  a  lens.  Such  a  group  of  slit  images  is  called 
a  line  spectrum.  This  separation  of  the  colors  is  called  dispersion. 
If  the  source  emits  waves  of  an  infinite  number  of  lengths  in- 
cluded between  the  red  and  the  violet,  the  infinite  number  of 
partially  overlapping  images  of  the  slit  will  form  a  continuous 
spectrum.  If  the  slit  is  wide  the  different  colors  will  greatly  over- 


REFRACTION  AND  DISPERSION  587 

lap,  and  the  spectrum  is  said  to  be  impure.  There  is  less  over- 
lapping when  the  slit  is  narrowed;  but,  since  no  slit  can  be  made 
infinitesimally  narrow,  it  is  manifestly  impossible  to  obtain  a 
perfectly  pure  spectrum. 

673.  Fraunhofer  Lines. — If  a  wide  slit  illuminated  by  sunlight 
is  used,  a  continuous  spectrum  is  observed,  apparently  like  that 
given  by  a  candle  flame.     Such  a  spectrum  was  observed  by 
Newton.     If,  however,  the  slit  is  very  narrow,  it  will  be  seen 
that  a  number  of  fine  dark  lines  parallel  to  the  slit  cross  the 
spectrum.     These  lines  were  first  seen  by  Wollaston  in  1802. 
He  observed  a  virtual  spectrum  by  looking  directly  through  a 
prism  at  an  illuminated  slit.     Fraunhofer,  about  1815,  by  the 
use   of   better   prisms,   and   by   forming  a  real  image  of  the 
spectrum  with  a  lens,  was  able  to  find  several  hundred  of  these 
lines,  which  are  now  usually  referred  to  as  Fraunhofer  lines.     It 
is  evident  from  this  that  the  solar  spectrum  differs  from  that  of  a 
candle  in  not  being  absolutely  continuous.     The  dark  gaps  in  1jhe 
position  of  different  colors  show  the  absence  of  corresponding 
images  of  the  slit,  and  therefore  the  absence  of  these  colors 
in  the  sunlight.     In  the  section  on  Absorption  it  will  be  shown 
that  most  of  these  dark  lines  are  due  to  the  absorption  of 
light  of  definite  wave  lengths  by  vapors  in  the  solar  atmosphere 
§737). 

The  Fraunhofer  lines  may  be  used  as  reference  points  in  meas- 
uring indices  of  refraction  of  prisms  for  different  colors.  The 
more  prominent  lines  were  labeled  by  Fraunhofer  with  letters  of 
the  Roman  and  Greek  alphabets.  Some  of  the  more  important 
of  them  are  the  A  line  (really  a  group  of  fine  lines) ,  due  to  absorp- 
tion by  the  earth's  atmosphere;  the  neighboring  D  lines,  due  to 
sodium  vapor  in  the  sun;  the  F  line,  due  to  hydrogen;  the  H  and 
K  lines,  due  to  calcium.  These  lines  are  shown  in  the  reproduc- 
tion of  the  solar  spectrum  (upper  part  of  Fig.  581). 

674.  Dispersive  Power. — The  deviation  of  a  particular  color 
by  a  prism  increases  with  the  index  of  refraction.     The  angular 
separation  or  dispersion  between  two  colors  depends  on  the  dif- 
ference between  their  respective   indices  of  refraction.     If   a 
prism  has  a  very  small  refracting  angle,  the  angles  of  incidence, 
refraction,  and  emergence  of  a  given  pencil  transmitted  at  the 
angle  of  minimum  deviation  will  likewise  be  small,  and  the  sines 


588 


LIGHT 


of   these    angles    may   be    considered    as    equal    to    the    arcs; 
consequently, 

sin  £  (A+D)     A+D 

sin  i  A  A 

Therefore 


If  Dl  and  D3  are  the  deviations  of  two  given  colors,  the  Fraun- 
hofer  lines  C  and  F,  for  example,  and  D2  that  of  an  intermediate 
color  halfway  C  to  Ft  D3  —  Dl  is  the  angular  dispersion  of  the 
extreme  colors  and  D2  is  the  mean  deviation  of  the  spectrum  of 
angular  width  D3  —  Z^.  The  dispersive  power  d  of  the  prism 
is  the  ratio  of  the  angular  dispersion  of  the  two  colors  to  their 
mean  deviation,  or 

,    D±-D>      (nt 


Newton  assumed  that  the  ratio  of  dispersion  between  two  given 
colors  to  the  mean  deviation,  or  the  dispersive  power,  is  the  same 


nv—nc 

HD 

UD  —  1 

Water  

I  .  3330 

0.0060 

0.0180 

CS-. 

1.6303 

.0345 

.0547 

Ether  

1.3566 

.0052 

.0149 

Alcohol 

1  3597 

.0062 

.0174 

Crown  glass  

1.5160 

.0073 

.0141 

Light  flint  glass 

1  5718 

.0113 

.0197 

Heavy  flint  glass  

1.7545 

.0274 

.0363 

Very  heavy  flint  glass 

1  .  9625 

.0488 

.0507 

Quartz  

1.5442 

.0078 

.0129 

Diamond     .         .       .           

2.4173 

.0254 

.0179 

2.1816 

.1256 

.1063 

Air  (0°  C.,  760  mm  )       

1  .  00024289 

.00000295 

.0121 

H,  

1.00014294 

.00000195 

.0136 

CO,. 

1  .  00044922 

.00000460 

.0102 

for  all  substances,  but  Dollond,  in  1757,  showed  that  this  is  by 
no  means  the  case.  Two  different  prisms  may  have  the  same 
value  of  n2 — 1,  but  very  different  values  for  n3  —  nlt  or  conversely. 


REFRACTION  AND  DISPERSION 


589 


The  table  shows  the  values  of  nD  and  the  dispersive  power 
between  the  C  and  F  lines  for  some  substances,  the  mean  devia- 
tion being  that  corresponding  to  the  D  lines.  There  are  great 
differences  between  the  refractive  and  dispersive  powers  of 
different  specimens  of  glass. 

675.  Irrational  Dispersion. — The  dispersive  power  of  a  given 
prism  (for  equal  increments  of  wave-length)  varies  in  different 
parts  of  the  spectrum,  usually  increasing  toward  the  violet. 
There  is  no  simple  ratio  between  deviation  and  wave-length, 
hence  such  spectra  are  said  to  be  irrational.  If  for  any  three 
colors  the  ratio  (n3  —  n^ /  (n2—  1)  were  the  same  for  all  substances 
the  spectra  formed  by  different  prisms  would  all  be  alike  in 
the  distribution  of  colors,  and  one  spectrum  would  be  simply  a 
larger  or  smaller  copy  of  any  other.  As  stated  above,  this  ratio 
is  not  the  same  for  different  substances,  so  that  the  spectra 
formed  by  different  prisms  are  also  irrational  with  respect 
to  each  other.  It  is  possible,  for  example,  to  make  a  prism  of 
crown  glass  and  one  of  flint  glass  which  will  give  spectra  of  equal 
length  between  the  lines  A  and  K;  but  it  will  be  found  that  the 
positions  of  the  other  Fraunhofer  lines  do  not  coincide  in  the 
two  spectra,  as  they  would  if  the  dispersion  were  rational. 

The  following  t;able  showing  the  differences  between  the 
refractive  indices  of  various  substances  for  the  A,  D,  F,  and  G 
Fraunhofer  lines  illustrates  irrationality  of  dispersion.  It  will 
be  seen  that  the  ratio  (nF  —  nD)  /  (nD  —  nA) ,  for  example,  is  not 
the  same  for  the  different  substances. 


np-nD 

« 

nD     nA 

UQ     Up 

*>D  —  nA 

Crown  glass 

0.00485 

0  00515 

0  00407 

1  062 

Heavy  flint  glass  
Water  

.01097 
.00409 

.01271 
.00415 

.01062 
00344 

1.158 
1  015 

CS, 

.01898 

02485 

02446 

1  309 

Although  as  a  general  rule  the  index  of  refraction  increases 
as  wave-length  diminishes,  there  are  exceptions,  as  described 
under  the  head  of  anomalous  dispersion  (§777). 


590 


LIGHT 


676.  Achromatic  and  Direct -vision  Prisms. — The  unequal 
dispersive  power  of  different  subst/ances  is  utilized  for  making 
prismatic  combinations  for  producing  deviation  with  very  little 
dispersion  (Fig.  519),  or  dispersion  without  deviation  of  the 
spectrum  as  a  whole  (Fig.  520).  These  two  types  are  respec- 
tively called  achromatic  and  direct-vision  prisms. 


FIQ    519. 


Fro,  520. 


LENSES 

677.  Lenses  are  transparent  bodies,  generally  with  spherical 
surfaces,  which  form  images  by  changing  the  divergence  of  light 
waves.     The  ordinary  types  of  single  lenses  are  shown  in  Fig. 
521.     The  first  three  forms,  known  as  double-convex,  plano- 
convex, and  concavo-convex,  are  thicker  at  the  center  than  at  the 
edges.     If  surrounded  by  a  less  refractive  medium,  the  central 
portion  of  the  incident  wave  is  more  retarded  than  the  edges  by 
these  lenses,  and  the  curvature  of  the  wave  is  diminished  or 
reversed  in  direction.     These  lenses  have,  therefore,  a  convergent 

effect .  They  are  called  convex 
or  converging  lenses.  In  the 
second  group,  embracing  the 
double-concave,  plano-con- 
cave, and  convexo-concave 
lenses,  the  edges  are  thicker 
than  the  center,  so  that  the 

outer  portions  of  an  incident  wave  are  more  retarded  than  the 
center.  The  curvature  of  the  wave  is  increased  and  the  lenses 
have  a  .divergent  effect.  Such  lenses  are  called  concave  or 
diverging  lenses.  If  the  two  types  of  lenses  are  placed  in  a 
more  refractive  medium  there  is  a  reversal  of  these  effects. 

678.  Equivalent  Air  Path  or  Reduced  Optical  Path. — At  a  given 
instant  a  wave  front  is  in  a  given  position ;  later  it  will  be  in  a  dif- 


Fio.  521. 


LENSES 


591 


ferent  position,  and  may  have  its  orientation  and  curvature 
greatly  modified  by  reflection  or  refraction.  The  one  condition 
that  must  always  be  fulfilled,  if  the  wave  is  to  preserve  its 
identity,  is  that  the  time  required  for  the  disturbance  to  travel 
from  a  point  in  the  original  wave  front  to  the  corresponding 
point  in  the  new  wave  front  is  the  same  for  all  parts  of  the  wave. 
For  example,  the  disturbance  traveling  from  S  by  the  path 
SPRQSl  (Fig.  522)  reaches  Sj^  at  the  same  time  as  the  disturb- 
ance leaving  S  at  the  same  instant  and  traveling  along  SAES^ 
The  latter  has  been  sufficiently  retarded  by  passing  through  a 
greater  thickness  of  glass  to  compensate  for  the  greater  dis- 
tance in  air  SPRQSt.  Similarly,  the  time  required  for  the 
wave  to  travel  from  P  to  Q  is  the  same  as  that  from  B  to 
D.  In  comparing  the  distances  traversed  in  equal  times  in 
different  media,  account  must  be  taken  of  the  velocity  of  light  in 
the  respective  media.  For  example,  in  Fig.  522,  PR  +  RQ  =  7^; 
BD  =  VJ.  Therefore,  PR+RQ  =  (VJV2)BD  =  nBD.  If  BD  is 
the  distance  actually  traversed  in  a  medium  of  refractive 
index  n,  the  equivalent  air  path  or  reduced  optical  path  is 
nBD. 

679.  Conjugate  Focal  Relations. — Consider  the  case  of  a  double- 
convex  lens  of  refractive  index  n  surrounded  by  air,  the  refractive 
index  of  which  may  be  taken  as  unity.  Let  the  radius  of  curva- 
ture of  the  first  surface  of  the  lens  be  rlt  that  of  the  second  r2. 


Pr      R       Q' 


FIQ.  522. 


Let  u  be  the  distance  of  the  source  from  the  lens.  PB  is  a 
section  of  the  incident  wave  front  of  radius  uy  and  QD  that  of 
the  emergent  wave  front,  of  radius  v. 

The^disturbance  actually  travels  radially  from  Pto  R,  thence  to 
Q,  but  if  a  is  very  small,  the  path  in  air  may  be  assumed  to  be 


592  LIGHT 

equal  to  P'Q'  without  appreciable  error.    Placing  the  optical  path 
through  the  center  of  the  lens  equal  to  this  distance,  we  have 


or 

AB  +  DE=(n-l) 

Substituting  reciprocal  radii  of  curvature  for  sagittse  (§658),  this 
becomes 


1,1,       -./I      1\      1 

-H —  (ft -1)1  —  +  —     =  ~.r 

u     v     v        Vi     V     / 


In  this  u,  v,  rly  and  r2  are  considered  as  mere  lengths,  that  is, 
numerical  quantities  without  sign.  As  we  shall  later  treat  them 
as  algebraical  quantities  with  signs,  it  may  be  noted  that,  in  the 
above  case,  u  and  v  are  both  measured  in  the  direction  in  which 
the  light  proceeds.  It  should  also  be  noticed  that  the  refraction 
at  the  first  surface  makes  the  wave  less  divergent,  that  is,  it  tends 
to  converge  it  toward  the  opposite  side.  The  same  is  true  of  the 
second  surface.  Hence  both  surfaces  may  be  described  as  con- 
verging surfaces.  If  the  curvature  of  either  surface  were 
opposite  to  its  direction  in  the  double-convex  lens,  it  would 
be  a  diverging  surface. 

If  the  source  of  light  is  at  an  infinite  distance,  that  is,  if  the 
incident  waves  are  plane,  u  =  oo  and  v  —f.  Hence  /  is  the  dis- 
tance of  the  point  called  the  principal  focus,  to  which  the  lens 
converges  plane  waves. 

In  the  case  of  a  double  concave  lens,  of  thickness  CD  along 
the  axis  (Fig.  523),  if  the  incident  wave  front  is  PB  and  the 
emergent  wave  front  QF,  put  the  optical  path  BF  equal  to  the 
optical  path  PQ  (assumed  parallel  to  the  axis,  since  a  is  small)  . 


.'.AB-EF=(n-l)  (-AC-DE) 
Substituting  radii  of  curvature  for  sagittse, 


1  1  ,  1N/  1  1\  1 
---  —  (n—  1)(  -----  )  —  —  -3T 
u  v  J\  r,  rj  f 


The  above  formula  would  be  identical  with  that  for  the  double 
convex  lens  if  the  signs  of  t>,  rlt  and  r,  were  reversed.     Now 


LENSES 


593 


each  of  these  quantities,  in  the  case  of  the  double  concave  lens, 
is  actually  measured  in  the  opposite  to  the  direction  in  which 
it  is* measured  in  the  double  convex  lens.  Hence,  if  we  take  the 
latter  direction  as  positive  for  each,  and  so  treat  all  of  them 
(and  v)  as  algebraical  quantities,  the  two  formulae  will  become 
identical. 

From  Fig.  523  it  is  evident  that  incident  light  is  made  more 
divergent  by  the  lens.  In  fact  both  surfaces  are  diverging  sur- 
faces. The  significance  of  the  negative  sign  of  /  in  the  above 
formula  is  that  the  principal  focus  is  virtual,  its  distance  from  the 
lens  being  measured  in  a  direction  opposite  to  that  in  which  the 
light  actually  travels. 


Fiq.  523. 

By  applying  the  same  method  to  the  other  types  of  spherical 
lenses  it  will  be  found  that  the  general  solution  of  all  cases  is  the 
formula 


1,1     f       .Jl   ,  1\     1 
-+-**(n—  1)   — —  I —-3 

*     v  Vi     r*J     ! 


provided  we  adopt  the  following  rule  regarding  signs: 

Consider  each  of  the  quantities,  u,  v,  rl}  r2,  /,  as  positive  when 

it  is  on  the  same  side  of  the  lens  as  in  the  typical  case  of  a  double 

convex  lens  forming  a  real  image — negative  when  on  the  opposite 

side. 

Negative  values  of  v  and/  indicate  that  the  light  diverges  from 


594  LIGHT 

a  virtual  focus  after  passing  through  the  lens.     These  conven- 
tions are  consistent  with  those  of  (§659). 
The  following  cases  arise  when /is  positive: 

When  u  =  oo ;  v  =/,  the  principal  focal  distance. 
When  u  >  f,  v  is  positive  and  there  is  a  real  conjugate  focus. 
When  u  —f,  v  =  oo .    The  transmitted  beam  is  parallel. 
When  u  <  f,  v  is  negative  and  greater  than  u  for  all  positive 
values  of  u,  and  there  is  a  virtual  conjugate  focus. 

680.  Axes  of  Lens. — The  line  passing  through  the  centers  of 
curvature  of  the  surfaces  of  a  lens  is  called  the  principal  axis. 
In  every  lens  there  is  a  point  on  the  principal  axis,  called  the 

optical  center,  which  has  the 
property  that  no  ray  passing 
through  it  is  deviated  in  direc- 
tion, although  there  is  more 
or  less  displacement,  depend- 
ing on  the  thickness  of  the 
lens. 

The  existence  of  this  point 
may    be    shown   thus:     Let 
Fia  524.  two  parallel  radii  of  curvature 

rt  and  r2  (Fig.  524)  be  drawn 

to  the  two  surfaces  of  a  lens.  Since  the  two  plane  elements  of 
the  lens  A  and  A'  are  parallel,  being  respectively  perpendicular 
to  two  parallel  lines,  the  refracted  ray  A  A'  is  propagated  in  a 
medium  with  parallel  sides  and  emerges  parallel  to  its  original 
direction.  Since  the  triangles  ACCl  and  A'CC2  are  similar, 


CCl    CC, 

This  is  true  whatever  may  be  the  value  of  the  angle  a,  there- 
fore C  is  a  fixed  point,  the  optical  center  of  the  lens.  All  ray 
paths  which  pass  through  this  point  are  called  secondary  axes. 
In  the  case  of  a  thin  lens,  the  center  of  the  lens  and  the  optical 
center  may  usually  be  regarded  as  coincident. 

681.  Images  by  Lens. — The  image  of  A  (Fig.  525,  a,  6,  c)  must 
lie  on  the  secondary  axis  A  A',  that  of  B  on  the  secondary  axis 
BB'.  Rays  drawn  parallel  to  the  principal  axis  from  the  points  A 
and  B  pass  throjugh  the  principal  focus  F  and  intersect  the  lines 


LENSES 


595 


A  A'  and  BB'  at  the  points  A'  and  B',  which  determine  the  posi- 
tion and  magnitude  of  the  image.  Since  the  point  A'  lies  above 
the  principal  axis  when  the  image  is  on  the  same  side  of  the  lens 
as  the  object,  and  below  it  when  the  image  is  on  the  other  side 
of  the  lens,  it  is  evident  that  all  virtual  images  formed  by  a  single 
lens  are  erect,  all  real  images  inverted. 


B1 


b 

FIQ.  525. 


Since  an  object  AB  and  its  image  A'B'  subtend  equal  angles 
from  the  center  of  the  lens  (the  angle  included  between  the  sec- 
ondary axes  A  A'  and  BB')  it  is  evident  that  their  relative  sizes 
are  proportional  to  their  respective  distances  from  the  lens,  or 


_ 
A'Bf 


Spherical  Aberration. — In  deriving  the  formula  for  the 
conjugate  focal  relations  of  lenses  it  has  been  tacitly  assumed  that 
the  emergent  wave  is  spherical. 
With  lenses  of  small  aperture 
this  is  shown  by  experience  to 
be  practically  true;  but  when 
the  aperture  becomes  large 
there  is  noticeable  spherical 
aberration.  This  is  illustrated 
by  Fig.  526. 

While  the  central  part  of  the 
wave  travels  from  B  to  C  the  edge  of  the  wave  will  travel  along 
LMNthe  distance  LMNO  =  PQ  =  nBC>LMN.  It  is  evident, 
therefore,  -that  the  edge  of  the  emergent  wave  (represented 
by  the  dotted  curve)  will  pass  through  0  instead  of  N,  and  will 


Fio    526. 


596 


LIGHT 


have  a  greater  curvature  toward  the  axis  than  if  the  wave  were 
spherical,  with  St  as  a  center.  The  rays,  instead  of  converg- 
ing to  Slt  as  shown  in  the  lower  half  of  Fig.  526,  will  cross  each 
other  as  shown  in  the  upper  half,  being  enveloped  by  a  caustic 
surface  instead  of  by  a  right  cone. 

683.  Correction  of  Spherical  Aberration. — If  the  rays  passing 
through  the  edge  of  a  lens  are  stopped  by  a  diaphragm  which 
permits  only  the  central  portion  of  the  incident  pencil  to  pass 
the  spherical  aberration  will  be  greatly  reduced.  It  is  also  pos- 
sible to  grind  surfaces  slightly  differing  from  a  spherical  form, 
so  that  for  a  given  pair  of  conjugate  focal  distances  the  emer- 
gent wave  is  truly  spherical.  Such  lenses  are  called  aplanatic. 
In  some  cases  when  the  conjugate  focal  distances  differ  greatly, 
spherical  aberration  may  be  reduced  by  making  the  two  surfaces 
of  the  lens  of  different  curvatures.  Consider,  for  example,  a 


FIG.  527- 


FIG.  528. 


plano-convex  lens  of  great  aperture  (Figs.  527,  528)  first  with 
the  plane  then  with  the  convex  face  toward  a  source  so  distant 
that  the  incident  light  is  parallel  or  nearly  so.  If  we  consider 
the  deviation  of  the  ray  PQ  in  each  case,  it  is  evident,  on  re- 
calling the  condition  for  minimum  deviation  by  a  prism,  that  in 
the  second  case  the  angle  D  will  be  less  than  in  the  first,  because 
the  refracting  edge  of  the  lens  is  then  more  nearly  in  the  position 
with  respect  to  the  incident  and  emergent  rays  which  gives  mini- 
mum deviation,  and  consequently  the  nearest  approach  of  the 
ray  PQ  to  the  focus  Sr 

One  form  of  thick  lens  of  great  angular  aperture  commonly  used  as  part 
of  microscopic  objectives  is  almost  entirely  free  from  spherical  aberration. 

Suppose  that  a  ray  of  light  which  starts  from  S  in  a  transparent  sphere 
of  radius  R  is  refracted  at  0  along  a  line  that  intersects  CS  produced  in  Sr 
We  shall  show  that,  if  CS=R/n,  then  CS^nR,  and  therefore  all  rays 
from  S  seem,  after  refraction,  to  come  from  one  point  Sr 


LENSES 


597 


Using  a  well-known  geometrical  principle  we  may  write 

OS  =  sin  rt    CSl==  sin  i 

/£  ~sin  a'     #    **  sin  /? 

sin  r     1     sin  r 

f*f    SS    —    S=B 

sin  a     n     sin  i 

.*.   a=i 
Since  the  angle  C  is  common  to  the  two  triangles  COSV  COS 


=  r 


Hence 


sin  % 
—  —  77 
sin/? 


sin  i 

/  -:  - 
sin  r 


This  is  an  exact  relation,  no  matter  how  large  the 

angle  i  may  be,  so  that  an  object  in  the  lens  at  S 

would  have  a  virtual  image  at  St  entirely  free 

from  aberration.     The  same  is  practically  true  if 

the  segment  of  the  sphere  below  S  is  removed 

and  the  object  placed  in  contact  with  the  sur- 
face.    In  practice  this  lens  is  often  a  hemisphere, 

the  object  being  placed  at  such  a  distance  below 

the  plane  side  that  its  virtual  image  formed  by 

refraction  at  the  plane  surface  corresponds  in  posi- 
tion to  the  point  S.     There  is  some  aberration  in 

this  case  due  to  refraction  at  the  lower  surface.     In 

the  method  of  "  oil  immersion"  the  object  is  placed 

at  S  and  the  space  between  it   and    the  hemi- 

spheral  lens  is  filled  in  with  an  oil  of  the  same 

refractive  index  as  the  lens. 

684.  Focal  Lines. — If   a  pencil  of  light  falls  obliquely  on  a  converging 

lens,  instead  of  a  point  image  two  real  focal  lines  will  be  formed,  like  those 

due  to  a  concave  mirror.     If  the  lens  is  divergent,  these  focal  lines  will  be 

virtual.     The  formation  of  these  lines  by  a  converging  surface  is  made  clear 

by  considering  the  effect  of  a  single 
refracting  surface  PQ,  imagining  it 
to  be  extended  to  R,  so  that  SSt  is 
the  principal  axis  (Fig.  530).  The 
rays  transmitted  through  the 
actual  refracting  surface  PQ,  will, 
by  reason  of  spherical  aberration, 
pass  through  a  narrow  arc  through 
This  is  the  primary  focal  line.  These  rays 


Fio.  529. 


Fio.  530. 


Fl  with  its  center  on  SSr 


will  all  intersect  the  axis  SSl  between  Sl  and  T.  The  normal  cross- 
section  of  this  pencil  is  a  narrow  lemniscate-shaped  region  at  F2,  the 
secondary  focal  line,  at  right  angles  to  the  primary  focal  line.  The  second 
refarcting  surface  of  the  lens  will  modify  but  not  change  the  gerersl 
character  of  this  result 


598 


LIGHT 


685.  Cylindrical  Lens. — The  effect  of  such  a  lens  is  like  that  of  a  cylin- 
drical mirror.  A  point  source  S  has  two  linear  imgess,  as  shown  in  Fig.  531, 
one  AB  parallel  and  the  other  A'B'  at  right  angles  to  the  axis  of  the  lens. 
The  image  AB  parallel  to  the  axis  is  at  a  distance  given  by  the  relation 


U       V 

and  may  be  either  real  or  virtual;  the  other, 
.  A'B'  is  virtual  and  may  be  considered  as 
due  to  each  longitudinal  strip  of  the  lens 
acting  as  a  prism  of  the  same  angle.  Any  lens 
with  different  curvatures  in  planes  at  right 
angles  to  each  other  will  give  similar  focal 
lines  or  astigmatic  images. 

686.   Combinations   of   Lenses. — If 
the  lenses  are  thin,  with  principal  focal 

lengths  /j  and  /2,  and  so  close  that  the  distance  between  them 
may  be  neglected, 

!+!„!._!+!=! 

u     w    /i ;       w      v      /2 

If  w,  the  focal  distance  conjugate  to  u,  is  positive  with  respect 
to  the  first  lens,  it  is  negative  with  respect  to  the  second. 
Therefore, 

M-7-H 

V          Jl        J2        J 

This  expression  is  generally  true  for  either  converging  or  diverg- 
ing lenses  if  the  proper  signs  are  given  to/t  and/2. 
687.  Chromatic  Aberration. — Since 

1 


(n-1)   f+^ 


it  is  evident  that  the  princi- 
pal focal  distances  are  differ- 
ent for  different  colors,  being 
less  for  violet  than  for  red 
(Fig.  532).  There  is  no  way 
to  remedy  this  defect  in  a  Fia-  532- 

single  lens,  but  it  may  be  greatly  reduced  by  a  suitable  combi- 
nation of  lenses. 

688.  Achromatic  Combinations. — By  combining  two  or  more 
lenses  of  different  dispersive  powers,  two  or  more  given  colors 


REFRACTION  PHENOMENA 


599 


may  be  brought  to  the  same  focus,  just  as  prisms  may  be  com- 
bined to  give  deviation  without  dispersion  (Fig.  533).  If  two 
lenses  are  used,  for  each  color 

i+i-i 
/i  /•  / 

if  the  lenses  are  in  contact. 
If  we  wish  to  combine  the 
two  colors  corresponding  to 
the  C  and  F  lines,  /  must  be  the  same  for  both. 

If  up'  and  ntf   be  the  refractive  indices  of  the  first  lens,  np"  and 
those  of  the  second, 


Fia.  533. 


therefore, 


Fio.  534. 


The  values  of  X1-l/r1/  +  l/ra/  and  Xj-l/r/'  +  l/r,"  may  be  arbitrarily 
chosen  to  satisfy  this  relation.  Since  np>nQ  it  is  evident  that  Kl  and  Kt 
must  be  of  opposite  sign,  so  that  either  fl  or  /2  must  be  negative  If  /, 
is  negative  and  greater  than  fv  f  is  positive  and  the  lens  is  convergent. 
If  /,  is  negative  and  less  than  /t  the  combination  is  divergent.  Usually 
the  positive  lens  is  of  crown  glass,  the  nega- 
tive  of  flint,  and  they  are  shaped  to  fit  close 
together,  so  that  r/— r/'  and  often  r2"  —  oo 
(Fig.  533). 

Chromatic  aberration  may  also  be  reduced 
by  using  two  lenses  of  the  same  index  of 
refraction  at  a  certain  distance  d  from  each 
other.  To  take  a  specific  case,  if  the  second  lens  is  placed  at  a  distance  from 
the  first  equal  to  its  own  focal  length,  the  rays  of  different  colors  which 
diverge  from  each  other  at  the  first  lens  will  be  made  approximately  parallel 
by  the  second.  If  an  object  is  placed  at  the  principal  focal  point  F  (Fig.  534) 
of  the  combination,  a  virtual  image  at  infinity  will  be  formed,  and,  as  shown 
by  the  figure,  the  violet  and  the  red  images  will  subtend  approximately 
equal  angles  a  at  the  eye,  and  will,  therefore,  be  superimposed  on  the  retina. 

REFRACTION  PHENOMENA 

689.  Total  Reflection. — If  a  ray  of  light  travels  from  a  more  to 
a  less  refractive  medium,  the  angle  of  emergence  i  is  greater  than 


600 


LIGHT 


the  angle  of  incidence  (which,  being  in  the  more  refractive 
medium,  may  still  be  called  r  for  consistency).  Since  sin  r  = 
(sin  t)/n,  and  since  i  has  a  maximum  limit  of  90  degrees,  r  has  a 
maximum  limit  k  such  that  sin  k  =  l/n.  No  pencil  incident  on 
the  boundary  at  a  greater  angle  than  k,  the  critical  angle,  can 
emerge.  It  will,  therefore,  be  totally  reflected  (CC",  Fig.  535). 
Since  sin  A;  varies  inversely  as  the  index  of  refraction,  the  critical 
angle  is  different  for  different  colors.  Violet  will  first  be  subject 
to  total  reflection  as  r  increases,  and  finally  the  red. 


flf 


Fio.  535. 

A  parallel-sided  plate  cannot  be  used  to  show  total  reflection, 
since  any  pencil  entering  such  a  plate  must  emerge  at  the  same 
angle.  Objects  of  prismatic  form  are  best  adapted  for  the  purpose. 
The  effect  may  be  seen  by  looking  down  through  the  side  of  a  glass 
containing  water  or  at  a  test-tube  sunk  in  water.  A  fish  can  see 
objects  throughout  the  space  above  the  water,  but  he  sees  them 
through  the  limited  cone  of  angle  2k  =  97°  (Fig.  536) ,  arranged 
around  a  circle,  with  tops  pointing  inward. 

Some  values  of  k  are  given  below: 


Water 48°  36' 

Crown  glass 43°  2' 

Flint  glass 37°  34' 


Quartz 40°  22' 

Diamond..  ..24°  26' 


The  smaller  the  critical  angle  of  a  jewel  with  regular  facets, 
the  greater  the  proportion  of  light  totally  reflected  by  it.  This 
explains  the  great  brilliancy  of  the  diamond. 

The  index  of  refraction  of  a  liquid  or  of  a  small  portion  of  an 
opaque  object  may  be  determined  by  measuring  the  angle 
of  total  reflection  from  its  surface  when  in  contact  with  a  more 
refractive  medium  and  using  the  relation  sin  k  =  n/nlf  where 
n  is  the  index  of  the  less  and  nt  that  of  the  more  refractive 
medium. 


REFRACTION  PHENOMENA  601 

690.  Transition  Layer.  —  It  seems  quite  possible  that  the  change  of  index 
of  refraction  at  the  boundary  is  not  abrupt,  but  that  there  is  a  transition 
layer  /  due  to  interpenetration  of  the  two  media,  or  occlusion  at  the  sur- 
face causing  a  gradual  change  in  the  index.  If  this  be  the  case,  total 
reflection  may  be  considered  as  altogether  due  to  refraction.  When  the 
angle  of  incidence  is  equal  to  or  greater  than  k  the  wave  front  in  the  transi- 
tion layer  will  swing  around  and  become  normal  to  the  surface  (Fig.  537)  ; 
then  the  lower  edge  will  gain  on  the  upper  and  the  wave  will  swing  back 
into  the  first  medium.  If  we  consider  an  air  film  between  two  refracting 
media  the  two  transition  layers  may  encroach  on  each  other  (Fig.  538), 
in  which  case  the  lower  edge  of  the  wave  will  be  retarded,  and  a  part  of  it 
will  pass  into  the  third  medium.  It  might  be  expected,  therefore,  that  if 


Fio.  537.  Fia.  538. 

the  air  film  from  which  total  reflection  takes  place  is  very  thin  total  reflection 
will  cease.  This  has  been  found  to  be  the  case.  If  a  right-angled  total 
reflecting  prism  with  a  slightly  convex  hypothenuse  surface  is  pressed 
against  a  glass  plate,  total  reflection  takes  place  from  the  hypothenuse  of 
the  prism  when  the  angle  of  incidence  i  is  sufficiently  large,  but  some  light 
will  always  be  transmitted  through  the  region  surrounding  the  point  of 
contact  even  where  the  air  film  has  a  measurable  thickness.  It  is  found  that 
the  thickness  of  the  air  film  through  which  transmission  can  occur  (which 
may  be  considered  as  approximately  the  thickness  of  the  transition  layer) 
differs  with  the  wave-length  and  with  the  angle  of  incidence,  and  may  reach 
several  thousandths  of  a  millimeter. 


Fio.  539. 

691.  Mirage. — Examples  of  the  type  of  total  reflection  referred 
to  above  are  found  in  the  case  of  refraction  by  gases  of  varying 
density.  This  phenomenon  is  called  mirage.  The  air  above  a 
furnace  or  a  heated  surface  such  as  a  pavement  exposed  to  the 
sun's  rays  rapidly  increases  in  density  and  refractive  power 
in  going  upward.  If  the  line  of  vision  forms  a  small  angle  with 


602  LIGHT 

the  surface,  distant  objects  are  seen  apparently  reflected  from 
the  surface.  The  formation  of  one  type  of  mirage  is  shown  in  Fig. 
539.  The  object  AB  is  viewed  directly  through  the  pencils  OA, 
OB,  while  an  inverted  image  A'B'  is  also  seen,  due  to  the  refraction 
of  the  pencils  OEA,  OFB,  by  the  heated  air  near  the  ground. 
This  is  one  of  several  types  of  atmospheric  mirage.  Other  types 
showing  distortion  or  displacement  of  objects  are  due  to  local  differ- 
ences of  temperature  in  the  atmosphere,  which  cause  changes  in 
density  and  refractive  power.  They  are  very  easily  seen  by 
viewing  obj  ects  at  a  grazing  angle  across  heated  surfaces.  Similar 
effects  are  to  be  seen  by  looking  through  sheets  of  glass  with 
irregular  surfaces,  or  non-homogeneous  mixtures  of  liquids,  such 
as  water  with  an  excess  of  salt  crystals  at  the  bottom  of  the 
vessel,  or  with  alcohol  above  and  imperfectly  mixed  with  it. 

When  the  sun  is  near  the  horizon,  the  rays  reaching  the  eye 
traverse  strata  of  air  of  gradually  increasing  density,  which  cause 
them  to  bend  downward.  For  this  reason  the  sun  is  visible 
when  it  is  actually  below  the  horizon  a  distance  about  equal  to 
its  own  diameter. 

The  scintillation  of  stars  is  due  to  a  similar  cause.  Their 
apparent  direction  and  intensity  are  subject  to  rapid  fluctuations 
as  masses  of  air  of  varying  density  drift  across  the  line  of  sight. 

If  sunlight  be  focused  on  a  small  hole  beyond  which  it  diverges  to  a 
screen,  a  jet  of  coal  gas,  hydrogen,  or  carbon  dioxide  will  cast  a  clear  image 

on  the  screen.  The  difference  between 
the  refractive  power  of  the  jet  and  that 
of  the  air  will  cause  an  alteration  in  the 
distribution  of  light  on  the  screen,  which 
will  make  the  projected  areas  lighter  or 
darker  than  the  surrounding  space. 
540>  Ether  vapor  poured  from  a  beaker,  or 

ether  vapor  vortex  rings,  may  thus  be 

made  visible.  If  a  Bunsen  burner  be  placed  in  the  cone  of  light  a  beau- 
tiful representation  of  the  flame  and  the  currents  of  heated  gases  will  be 
formed  on  the  screen. 

692.  The  Rainbow  is  a  bright  arc  showing  the  spectral  colors, 
due  to  the  sunlight  refracted  by  raindrops.  Sometimes  several 
bows  are  seen,  the  inner  or  primary  ^bow  being  always  the 
brightest,  and  all  being  arcs  of  circles  with  centers  on  the  prolon- 
gation of  the  line  passing  from  the  sun  through  the  observer. 


REFRACTION  PHENOMENA 


603 


The  primary  bow  is  violet  on  the  inside,  red  on  the  outside;  in 
the  secondary  bow  the  order  of  colors  is  reversed. 

If  parallel  rays  are  incident  on  the  upper  half  of  a  refracting 
sphere  they  will  be  in  part  refracted,  internally  reflected,  and 
transmitted  downward  as  shown  in  Fig.  541 .  Rays  will  also  enter 
the  lower  half,  and  there  will  be  multiple  reflection  within  the 
sphere,  but  for  the  present  we  shall  fix  our  attention  upon  the 
rays  which  reach  the  eye  at  0  after  one  internal  reflection.  As 
indicated  by  the  course  of  the  rays  incident  at  P,  Plt  and  P2, 
there  is  an  angle  of  minimum  deviation,  below  which  no  rays 
once  internally  reflected  pass.  All  the  rays  emerging  at  nearly 
this  angle  are  parallel  or  nearly  so,  and  therefore  their  intensity 
varies  little  with  the  distance  from  the  drop,  while  rays  emerg- 
ing in  other  directions  are  widely  divergent. 


Fio.  541. 


Fia.  542. 


As  n  varies  with  the  color,  the  minimum  deviations  are  differ- 
ent for  the  various  colors.  In  the  primary  bow  the  minimum 
deviation  of  the  red  is  137°  42';  of  the  violet,  139°  37'.  In  the 
secondary  bow  the  corresponding  angles  are  230°  34'  and  233°  56'. 

From  Fig.  541  it  appears  that  light  will  be  received  by  the 
observer  at  0  from  all  the  raindrops  lying  in  an  arc  subtending  an 
angle  180° — D  with  the  axis  passing  from  the  sun  through  the 
observer's  eye.  In  the  primary  bow  this  angle  is  42°  18'  for  the 
red  and  40°  23'  for  the  violet,  so  that  the  bow  will  be  bordered  with 
violet  on  the  lower  side,  red  on  the  upper.  The  secondary  bow  is 
due  to  rays  incident  on  the  lower  half  of  the  drop,  twice  internally 
reflected,  and  then  transmitted  downward,  thus  inverting  the 


604 


LIGHT 


order  of  the   colors  (Fig.  542).     The  angle  subtended  by  this 
bow  is  D— 180°,  or  50°  34'  for  the  red,  53°  56'  for  the  violet. 

An  artificial  rainbow  may  be  made  by  causing  a  beam  of  sunlight  to  fall 
on  a  spherical  vessel  filled  with  water,  through  an  opening  in  a  screen.  The 
interior  of  the  circle  reflected  on  the  screen  is  illuminated  by  the  scattered 
light  which  has  been  once  reflected,  while  the  space  between  the  primary 
and  secondary  bow  is  quite  dark. 


INTERFERENCE. 


Examples  of  Interference.— Fresnel,  a  young  French  artil- 
lery officer,  about  1815,  produced  effects  similar  to  those  described 

in  §648  by  light  diverging 
.S  from  a  slit  S  and  reflected 
from  two  adjacent  mirrors 
MM  inclined  at  such  a  great 
angle  so  as  to  be  almost  in 
the  same  plane.  As  shown 
by  Fig.  543,  the  light  arriving 
at  any  point  P  where  the 
pencils  overlap  appears  to  come  from  the  two  virtual  sources  S1 
and  £2,  the  effect  of  which  is  precisely  the  same  as  that  of  the 
two  real  sources  in  Young's 
experiment.  (The  term  vir-  8 
tual  source  applies  to  a  point  „ 
from  which  the  waves  appear 
to  diverge  without  really  orig- 
inating at  that  point) . 


FIG.  543. 


Fio.   544. 


Another  method  of  obtaining  similar  interference  effects  is  by  means  of  a 
convex  lens  L  cut  along  a  diameter  in  two  halves  which  are  slightly  separated, 

giving  two  real  or  virtual  images  of 
the  source,  from  which  the  waves 
diverge  and  overlap.  This  is 
known  as  the  Billet  split  lens  (Fig. 
545). 

The  interference  effects  due  to 
Lloyd's   single  mirror  (Fig.  546), 

are  caused  by  waves  coming  respectively  from  the  real  source  S  and  the 
virtual  source  Si.  The  fringes  are  easily  obtained  by  reflecting  the  light 
from  a  narrow  slit  or  lamp  filament  at  grazing  incidence  from  a  mirror  of 


FIG.  545. 


INTERFERENCE  605 

black  glass,  in  order  that  the  effects  may  not  be  complicated  by  reflection 
from  the  rear  surface. 

Fresnel  also  produced  interference  effects  by  the  use  of  a  biprism  B 
equivalent  to  two  prisms  of  very  small  refracting  angle  placed  base  to  base 
(Fig.  544).  Here,  again,  it  is  evident  that  the  transmitted  light  appears  to 
come  from  the  two  virtual  sources  Sl  and  Sa. 


FIG.  546. 

694.  Newton's  Rings. — Robert  Boyle  described  the  brilliant 
colors  observed  in  soap  bubbles  and  other  thin  films,  an  effect 
which  appeared  to  depend  solely  on  the  thickness  of  the  films, 
not  on  their  nature.  Newton  investigated  this  phenomenon, 
which  he  tried,  with  poor  success,  to  explain  in  terms  of  the 
emission  theory.  In  order  to  secure  a  thin  film  of  air,  varying 
in  thickness  in  a  determinate  manner  from  point  to  point,  he 
pressed  a  convex  glass  lens  of  great  radius  of  curvature  against  a 
piece  of  plane  glass.  If  light  falls  normally  on  such  a  combina- 
tion, light  of  a  given  color  is  found  to  be  reflected  in  a  greater 
proportion  than  the  other  colors  from  all  points  where  the  film 
has  a  given  thickness,  the  predominant  color  varying  with  the 
thickness.  As  the  loci  of  points  of  equal  thickness  form  circles 
about  the  region  of  contact,  colored  rings  are  observed  concentric 
with  this  point.  These  have  been  called  Newton's  rings,  or  the 
colors  of  thin  plates.  Colored  rings  are  likewise  observed  in  the 
transmitted  light.  These  are  not  so  brilliant,  however,  as  those 
due  to  the  reflected  light,  as  the  transmitted  colors  are  mixed 
with  a  large  proportion  of  unmodified  white  light.  The  colors 
in  the  two  sets  of  rings  are  complementary — that  is  to  say,  the 
light  transmitted  through  a  given  point  is  white  deprived  of  the 
color  which  is  most  strongly  reflected  from  that  point.  If 
monochromatic  light  is  used  the  rings  are  alternately  dark  and 
of  the  color  used.  In  a  wedge-shaped  film  these  bands  are  par- 
allel to  the  edge  of  the  wedge;  in  a  film  of  uniform  thickness 
circular  bands  are  produced  under  certain  conditions,  uniform 
color  effects  under  others.  These  colors  of  thin  plates  are  seen 
in  all  kinds  of  thin  transparent  films,  such  as  soap  bubbles,  films 
of  oil  on  water,  and  thin  sheets  of  mica. 


606  LIGHT 

695.  Explanation  of  Newton's  Rings. — Thomas  Young  showed 
that  the  colors  of  thin  films  can  be  very  simply  explained  as  a 
result  of  the  interference  of  waves  reflected  from  the  two  surfaces 
of  the  film,  as  shown  below. 

It  has  been  found  impossible  to  produce  interference  effects 
between  two  pencils  from  separate  sources,  or  from  different 
points  of  the  same  source.  There  is  no  permanent  concordance 
in  phase  relations,  amplitude,  or  direction  of  vibration.  We 
need,  therefore,  consider  only  one  point  of  the  source  at  a  time. 

The  effects  of  adjacent  points  will  be 
simply  superimposed  on  those  of  the 
first,  without  mutual  interference. 

Let  LM  and  NO  (Fig.  547)  repre- 
sent two  opposite  elements  of  surface 
of  the  air  film  producing  Newton's 
rings,  slightly  inclined  to  each  other 
and  so  small  that  they  may  be  con- 
sidered plane.  A  narrow  pencil  from 
the  point  S  of  an  extended  source  is 
incident  at  A,  where  a  small  part  is 
reflected,  the  remainder  being  trans- 
mitted to  B,  where  it  is  again  subject 
to  reflection  and  refraction.  This 

process  is  repeated  at  C,  D,  E,  etc.,  but  the  components  become 
very  weak  after  a  few  reflections.  Owing  to  the  inclination 
between  the  surfaces,  the  reflected  pencils  are  not  parallel,  but 
intersect  in  the  neighborhood  of  Plt  while  the  transmitted  com- 
ponents appear  to  diverge  from  P2.  If  the  film  increased  in 
thickness  toward  the  right,  these  points  of  intersection  would  be 
respectively  on  the  opposite  side  of  the  film.  The  reflected  and 
transmitted  pencils  may  respectively  be  brought  together  again 
by  the  lenses  L±  and  I/2,  at  the  points  Sl  and  S2.  A  maximum 
or  a  minimum  of  intensity  may  be  produced  at  these  points, 
according  to  the  phase  differences  between  the  component  rays. 
The  eye  or  observing  telescope  must  be  focused  on  Pt  or  P2  to 
get  the  most  distinct  effect.  If  the  film  is  very  thin  these  points 
practically  lie  at  its  surface.  If  either  the  thickness  of  the  film 
or  the  angle  between  its  surfaces  is  large  the  pencils  are  so 
distant  from  each  other  or  so.  divergent  that  they  cannot  all 


INTERFERENCE 


607 


enter  the  pupil  of  the  eye. 
become  indistinct  or  vanish. 


Under  such  conditions  the  bands 


696.  General  Expression  for  the  Difference  of  Path. — Let  n  be  the  index 
of  refraction  of  the  film,  ni,  that  of  the  surrounding  medium,  and  for  sim- 
plicity imagine  the  two  surfaces  to  be  parallel.  To  reach  the  wave  front  CP 
(Fig.  548)  the  light  reflected  from  A  travels  the  distance  AP  in  the  first 
medium  while  the  interfering  component  has  to  travel  the  distance  AB+BC 
in  the  film.  The  wave  would  travel  a  distance  n\AP  in  air  while  traveling 
the  distance  AP  in  the  medium  of  index  of  refraction  HI  and  the  distance 
n(AB+BC)  in  the  air  while  traveling  the 
distance  AB+BC  in  the  film  (§678).  Hence 
the  equivalent  difference  of  path  in  air,  or 
the  optical  difference  of  path  is  \£. 

"  / 


d=n(AB+BC)  - 
But  AC  sin  i=AP;  AC  sin  r  =  CQ;  therefore 


Fio.  548. 


hence 

d=n(AB+BC-CQ)  =n(AB+BQ) 
=nAR  cos  r—2nt  cos  r. 

If  the  film  is  of  a'ir,  n—  1,  d  =  2t  cos  i  (if  t 
is  the  angle  of  refraction  in  air) .  The  effects 
at  S\  and  Ss  are  due  to  the  superposition  of  all  the  components  arising  from 
multiple  reflections  within  the  film.  Between  successive  pairs  there  is  the 
same  phase  difference. 

697.  Phase  Changes  in  Reflection.— (See  §252.)  Evidently,  so  far  as 
geometrical  differences  of  path  are  concerned,  there  should  be  reenforce- 
ment  from  the  components  reflected  from  the  region  of  contact,  where  the 
thickness  of  the  film  is  so  small  compared  with  the  wave-length  of  light  that 
it  may  be  ignored.  As  a  matter  of  fact,  the  center  of  the  reflected  system 
of  fringes  is  black.  Young  inferred  by  analogy  that  at  the  boundaries  of 
different  media  light  waves  are  subject  to  changes  of  phase  similar  to  those 
observed  in  the  case  of  material  waves  §(252)  so  that  waves  incident  from 
air  on  a  more  refracting  medium  may  behave  like  waves  of  sound  reflected 
from  a  medium  denser  than  air,  while  a  light  wave  traveling  in  the  opposite 
direction  will  behave  like  sound  waves  emerging  from  the  free  end  of  an 
organ  pipe.  The  waves  reflected  from  the  upper  surface  of  the  air  film  pass 
from  a  more  to  a  less  refractive  medium ;  at  the  lower  surface  the  contrary 
is  the  case.  If  t  is  small  compared  with  the  wave-length,  there  should  be 
a  difference  of  half  a  period  introduced  hi  the  act  of  reflection,  which  will 
cause  destructive  interference.  The  transmitted  components  have  a  differ- 
ence of  phase  of  an  entire  period  caused  by  two  internal  reflections,  and 
therefore  will  be  concordant.  This  would  explain  the  black  spot  seen  in 
the  center  of  the  reflected  system  of  Newton's  rings.  It  is  also  observed 


608  LIGHT 

that  soap  films  as  they  get  thinner  run  through  a  brilliant  series  of  colors 
when  viewed  by  reflected  light,  finally  becoming  black  just  before  they  break. 

If  the  lens  is  of  crown  glass,  the  plate  of  flint  glass,  and  if  the  interspace  is 
filled  with  a  liquid  of  intermediate  index  of  refraction,  such  as  oil  of  cloves, 
the  central  spot  of  the  reflected  system  will  be  bright,  that  of  the  transmitted 
system  dark.  This  confirms  Young's  theory. 

When  Newton's  rings  are  produced  by  an  air  film,  the  condition  for  a 
maximum  of  given  wave-length  X  in  the  reflected  light  is  (remembering 
that  a  loss  of  half  a  period  in  reflection  is  equivalent  to  a  path  difference  of 
X/2), 


2t  cos  i 

A 

and  for  a  minimum, 

nX  =  2t  cos  t 

In  the  transmitted  light  the  maxima  are  given  by 

n\  =  2t  cost 
and  the  minima  by 


In  the  above  expressions  n  is  the  ordinal  number  of  the  rings  counted 
from  the  center. 

698.  Film  with  Parallel  Sides. — If  the  surfaces  of  a  thin'plate  are  perfectly 
plane  and  parallel  the  interfering  rays  are  parallel,  as  shown  in  Fig.  548, 
and  the  eye  or  observing  telescope  must  be  focused  for  infinity  to  see  the 
bands  clearly.    Since  the  difference  of  path  between  the  components, 
2nt  cos  t,  varies  with  the  angle  of  incidence,  the  phase  relations  will  be  dif- 
ferent for  rays  reflected  from  different  parts  of  the  film,  but  will  be  the 
same  for  all  rays  reflected  from  the  film  at  the  same  angle.     The  light  from 
every  point  Si,  Sz,  etc.,  of  an  extended  source  will  be  brought  to  separate 
points,  Sif,  Ss,  etc.,  on  the  retina,  so  that  there  will  be  no  overlapping  of 
effects.     The  bands  will  in  general  be  curved,  their  loci  being  given  by 
cos  t  =  constant.    Such  bands  are  sometimes  known  as  Haidinger's  fringes, 
or  fringes  of  equal  inclination. 

699.  Interference   by   Thick  Plates. — It  is   usually  impossible    to   get 
nterference  effects  by  the  use  of  a  single  wedge-shaped  plate  unless  the 

inclination  of  the  surfaces  is  very  slight,  because  the  interfering  pencils  will 
otherwise  be  too  divergent  to  enter  the  eye  simultaneously.  If  the  surfaces 
of  the  plate  are  perfectly  plane  and  parallel  it  is  easy  to  obtain  interference 
effects  with  monochromatic  light  with  great  differences  of  path  between 
the  components.  The  limit  to  the  possible  differences  of  path  which  may 
exist  seems  to  be  due  to  the  lack  of  perfect  homogeneity  in  the  light  from 
available  sources,  or  to  the  probable  fact  that  radiating  centers  emit  detached 
wave  groups  corresponding  to  successive  stimuli,  these  groups  having 
different  relations  of  phase,  amplitude,  and  the  direction  of  vibration,  so  that 
waves  of  one  group  cannot  interfere  with  those  of  another.  Consequently 


DIFFRACTION  609 

the  maximum  difference  of  path  which  can  exist  cannot  exceed  the  length 
of  such  a  train  of  waves. 

700.  Stationary  Light  Waves. — Stationary  waves  (§253)  may  be  ex- 
pected if  plane  waves  of  light  are  reflected  normally  from  a  mirror,  but  as 
the  distance  between  the  nodal  planes  is  only  A/2  and  light  waves  are  very 
short,  it  is  difficult  to  verify  their  existence.  Wiener  did  so  by  a  very  ingeni- 
ous device.  A  glass  plate  AB  (Fig. 
549)  was  covered  with  a  very  thin 
photographically  sensitive  collo- 
dion film,  and  placed  film  down- 

ward  over  a  silvered  mirror  MN       ||||||||ll,      ij|||||||||]||||||||      11111111111111!)    II* 
with  a  very  slight  inclination  be  Fio.  549. 

tween  the  two  surfaces.      After 

exposing  the  plate  to  a  beam  of  light  incident  normally  on  the  mirror  it 
was  developed,  and  dark  bands  were  found  in  the  film,  running  parallel 
to  the  line  of  intersection  of  the  two  surfaces,  as  indicated  by  the  shading 
below  M  N.  From  the  figure  it  is  clear  that  the  sensitive  surface  crossed 
the  nodal  and  anti-nodal  planes  in  such  a  manner  as  to  produce  this  effect. 
At  the  points  in  contact  with  the  mirror  no  effect  was  produced  in  the  film. 
This  proved  the  existence  of  a  nodal  plane  at  that  surface,  as  in  the  case 
of  the  analogous  sound  experiment.  This  is  the  basis  of  a  system  of  color 
photography  invented  by  Lippmann. 


DIFFRACTION 

701.  If  light  from  a  small  source  or  aperture  passes  by  the  edge 
of  an  obstacle  and  falls  on  a  screen  it  is  found  that  the  illumina- 
tion gradually  fades  away  in  the  geometrical  shadow,  while  out- 
side the  shadow  a  series  of  colored  bands  appears.  If  a  card  or 
knife  blade  is  held  between  the  eye  and  a  distant  source  of  light 
it  will  be  found  that  the  red  light  is  most  deflected  into  the  shadow, 
the  violet  the  least,  so  that  a  short  spectrum  is  formed.  Such 
phenomena  are  examples  of  what  is  known  as  Diffraction.  They 
are,  in  fact,  interference  phenomena  between  wavelets  coming 
from  adjacent  points  of  the  same  wave  front. 

Let  us  find  the  effect  of  an  extended  plane  wave  front  AB  (Fig. 
550)  at  the  point  P.  In  accordance  with  Huyghens'  principle, 
the  resultant  effect  at  P  may  be  regarded  as  the  sum  of  the  effects 
separately  due  to  all  the  points  in  the  wave  front,  each  originating 
its  independent  set  of  wavelets.  Waves  of  different  lengths  must 
be  separately  considered  in  this  analysis.  If  OP  =  r,  describe 
about  P  as  a  center  spheres  of  radii  r-|-X/2,  r+X,  r+3X/2,  etc. 

30 


610 


LIGHT 


These  spheres  will  intersect  the  wave  front  in  circles,  as  shown  in 
Fig.  551,  concentric  with  0,  the  pole  of  the  wave  with  respect  to  P. 
The  areas  between  successive  circles  are  called  half-period  zones. 


Fio.   550. 

The  student  may  easily  find  by  calculation  that  these  areas  are 
approximately  equal.  It  is  evident  that  the  disturbances  origi- 
nating in  all  points  in  a  circle  about  0  will 
reach  P  at  the  same  time,  and  that  the 
average  phases  of  the  resultant  effects  at 
P  of  successive  zones  will  differ  by  half  a 
period.  Although  the  areas  of  the  zones 
are  practically  the  same,  the  amplitude 
produced  by  each  at  P  slowly  diminishes  as 
its  radius  increases,  on  account  of  increas- 
ing obliquity  and  distance  with  respect  to 
P.  The  total  amplitude  produced  by  the 
wave  at  P  is,  therefore,  the  algebraic  sum  of  a  series  of  terms 
slowly  diminishing  in  magnitude  and  alternating  in  sign  (direc- 
tion of  displacement).  If  01,  #2,  «3,  etc.,  are  the  amplitudes 
at  P  due  to  the  central  area  and  successive  zones,  and  A  the 
resultant  amplitude. 


FIQ.  551. 


DIFFRACTION 


611 


As  the  successive  terms  differ  very  slightly  from  each  other  and 
diminish  in  accordance  with  a  regular  law,  the  quantities  in 
parentheses  and  an  may  each  be  placed  equal  to  zero.  Therefore, 


or  the  amplitude  at  P  due  to  the  whole  wave  is  one-half  and  the 
intensity  one-fourth  that  due  to  the  central  element  if  it  alone 
were  effective.  If  the  whole  wave  except  the  central  element 


Fia.  552. 

is  covered  the  illumination  at  P.  will  be  actually  increased,  the 
amplitude  in  that  case  being  ar  If  all  but  the  two  central  ele- 
ments are  covered  the  effect  at  P  is  A=al  —  a3  =  0  nearly.  If 
three  elements  are  uncovered,  A  =  al  —  a2  +  as  =  al  nearly.  These 
conclusions  are  easily  verified  by  experiment.  If  small  circular 
openings  of  different  sizes  be  placed  in  a  pencil  of  light  diverging 
from  a  pinhole,  maxima  and  minima  will  be  found  in  the  centers 


612 


LIGHT 


of  the  bright  areas  projected  on  a  screen  through  the  openings 
(Fig.  552) .  These  holes  decrease  regularly  in  size  from  1  to  9.  If 
the  screen  be  moved  (thus  changing  the  number  of  effective  half- 
period  elements  subtended  by  the  holes  at  the  screen)  maxima 
change  to  minima  and  vice  versa,  or  if  white  light  is  used  the 
bright  spot  at  the  center  changes  color.  The  central  spot  is  sur- 
rounded by  a  series  of  colored  bands  of  similar  origin,  but  not 
so  easily  explained  by  elementary  methods.  If  a  hole  is  smaller 
than  the  first  two  half-period  elements,  there  are  no  maxima  and 
minima  within  the  illuminated  area  on  the  screen,  as  there  can  be 
no  possible  disordance  of  phase  in  the  wavelets  coming  through 
the  hole,  and  consequently  a  diffuse  circular  patch  of  light  is  cast 

on  the  screen,  which  increases  in  size  as 
the  opening  is  made  smaller. 

If  a  small  disk  be  placed  in  the  path 
of  the  light,  so  as  to  cover  a  few  half- 
period  elements  as  viewed  from  P — say 
three — the  amplitude  at  P  will  be  A  = 
a4~ a5+a6~a7'"==ia4-  A  bright  spot 
will  therefore  be  seen  at  the  center  of 
the  shadow,  nearly  as  intense  as  though 
the  disk  were  removed.  At  adjacent 
points  not  on  the  axis  there  will  be  dis- 
cordance of  phase  between  the  disturbances  coming  around  the 
edge  of  the  disk,  resulting  in  destructive  interference. 

It  is  easy  to  perform  this  experiment  by  mounting  a  perfectly  circular 
disk  several  millimeters  in  diameter  on  a  piece  of  glass  plate  and  placing  it 
in  the  pencil  of  sunlight  from  a  small  pinhole  opening 
several  meters  away.  The  bright  spot  in  the  center 
of  the  shadow  may  then  be  seen  on  a  screen  a  few 
meters  beyond  the  disk.  A  reproduction  of  a  pho- 
tograph of  this  effect  is  shown  in  Fig.  553. 

702.  Waves  through  Large  Opening. — The  points 
a,  6,  c,  etc.,  in  the  wave  front  AB  (Fig.  554)  act 
as  centers  of  disturbance  and  propagate  wavelets 
to  the  tangent  surface  A'B'.  It  is  evident  from  the 
figure  that  only  those  wavelets  between  the  lines  A  A'  and  BB '  conspire  in 
a  common  wave  front.  Oustide  of  these  lines  the  wavelets  cross  each  other 
in  all  directions  and  in  all  possible  phases  at  random,  so  that  the  resultant 
disturbance  is  zero  except  in  the  immediate  neighborhood  of  A  A'  and  BB', 
where  diffraction  effects  are  produced. 


FIG.  553. 


Fio.  554. 


DIFFRACTION 


613 


B 


703.  Waves  through  Small  Opening. — From  Fig.  555  it  is  clear  that  no 
opposition  of  phase  between  the  elementary  wavelets  from  a,  6,  c,  etc., 
can  arise  until  the  point  Pl  is  reached,  where  the  difference  between  APl 
and  BPl  is  half  a  wave-length,  and  even  then  the 

disturbances  from  the  extreme  points  A  and  B  alone 
are  in  opposite  phase.  Only  when  this  difference  of 
path  is  a  whole  wave-length  can  complete  destruc- 
tion arise.  In  this  case  we  see  that  the  disturb- 
ances from  two  halves  of  the  opening  reach  Pl  with 
an  average  difference  of  path  of  half  a  wave-length, 
so  that  the  wavelets  nullify  each  other  pair  by  pair. 
If  the  opening  is  less  than  a  wave-length  in  width 
some  effect  is  produced  even  at  the  point  P3.  The 
effect  is  evidently  always  greatest  at  P0,  where  the 
wavelets  meet  very  nearly  in  the  same  phase,  and  least 
at  P,  where  there  is  the  greatest  diversity  of  phases.  Flo  555 

704.  Narrow  Slit. — If  two  straight  edges  are  opposed  so  as  to 
form  a  narrow  slit  of  width  AB  (Fig.  546)  there  will  be  a  bright 
band  at  P0  if  only  the  two  central  half-period  elements  of  an 
incident  wave  are  exposed.     If  two  on  each  side  are  exposed  the 
effect  at  P0  is 

A  =  2at  —  2a2  (nearly  zero) 

If  three  half-period  elements  on  each  side  are  exposed 
A  =  2al  —  2a2 + 2a3  (maximum) 

Thus  there  will  be  successive  maxima  and  minima  at  P0  as  the 

slit  is  widened.  If  the  slit  subtends  two 
or  any  even  number  of  half-period  ele- 
ments as  viewed  from  Plf  apoint  off  the 
axis  (T  being  its  pole),  they  will  neu- 
tralize each  other  in  pairs;  if  it  subtends 
an  odd  number  of  such  elements,  there 
will  be  destructive  interference  between 
pairs,  leaving  the  odd  one  effective, 

consequently  there  will  be  a  series  of  maxima  and  minima  on 

each  side  of  the  axis. 

706.  Width  of  the  Bands  Formed  by  a  Narrow  Slit—  If  AP.-BP^X 
(Fig.  557)  we  may  consider  the  effects  at  Pl  of  AO  and  OB  to  be  nearly 
the  same  numerically,  but  to  differ  in  average  phase  by  half  a  wave-length. 
The  two  cancel  each  other.  At  P2,  where  AP,— 5P2  «=f  ;,  we  may  imagine 
the  slit  divided  into  three  nearly  equal  strips,  which  contribute  effects  at  P 


Fio.  556. 


614 


LIGHT 


alternating  in  phase.  Two  cancel  each  other,  leaving  the  third  effective. 
If  D  is  the  distance  of  the  screen  from  the  slit,  the  width  of  the  central 
maximum  is  2P9Plt  and  it  can  easily  be  shown  as  in  §648  that 


The  other  bands  are  of  half  this  width,  or  D.X/AB.     The  width  of  all  the 
bands  is,  therefore,  inversely  proportional  to  the  width  of  the  slit.     The 


Fio.  557. 

central  maximum  is  white,  the  other  are  narrow  spectra  bordered  by  violet 
on  the  inside,  red  on  the  outside  (since  P0Pt  for  violet  is  less  than  the 
corresponding  distance  for  red.)  These  effects  may  be  observed  by  allowing 
light  from  a  narrow  slit  to  pass  through  a  second  adjustable  slit  and  fall 
on  a  screen,  or  more  simply  by  looking  through  a  narrow  slit  or  the  space 
between  two  fingers  at  a  distant  light. 


FIG.  558. 

Within  and  close  outside  the  shadow  of  a  wire  or  needle  cast  by  a  linear 
aource  similar  fringes  are  observed.  (See  Fig.  558,  showing  shadows  of 
needles  of  different  sizes.) 

706.  Resolving  Power. — If  the  light  from  a  narrow  slit  passes  through 
another  slit  to  a  screen  the  central  maximum  may  be  regarded  as  an  image 


DIFFRACTION  615 

of  the  first  slit  (corresponding  to  a  pin-hole  image).  The  wider  the  second 
slit  is  opened  (up  to  the  point  where  diffraction  effects  cease)  the  narrower 
and  sharper  this  image  will  be.  Similar  considerations  apply  to  light  from 
point  sources  through  circular  openings.  If  we  look  through  a  small  pin- 
hole  at  a  distant  light  it  will  appear  much  larger  than  when  viewed  with 
the  naked  eye.  The  filament  of  a  lamp  appears  thicker  when  seen  through 
a  narrow  slit.  If  an  image  is  formed  by  a  lens  or  mirror  the  same  conditions 
hold  as  for  a  narrow  slit,  the  lens  or  mirror  preserving  the  uniformity  of 
phase  of  the  whole  wave  with  respect  to  the  focus  that  exists  for  a  narrow 
slit  with  respect  to  its  central  maximum.  Consider  the  image  of  a  narrow 
source  S  (Fig.  559).  At  PI  and  P2  on  each  side  of  P0  there  will  be  a  minimum 


if  4P2-.BP2  =  X=BPi—  APi  in  which  case  the  disturbances  from  the  two 
halves  of  the  lens  reach  PI  and  P2  in  opposite  phases  and  cancel  each  other. 
The  width  of  the  image,  which  is  merely  a  diffraction  maximum,  is,  therefore, 
PiP2.  If  the  source  is  a  point,  the  central  maximum  will  be  of  the  shape  of 
the  opening,  but  differently  oriented,  because  any  particular  dimension  in 
the  source  will  be  inversely  proportional  to  the  same  dimension  in  the  open- 
ing, as  appears  from  the  relation  (§705). 


Observation  shows  that  two  diffraction  maxima  cannot  be  clearly  sepa- 
rated if  they  are  closer  than  the  distance  from  a  maximum  to  the  adjacent 
minimum.  If  the  image  of  S  for  example  lies  at  PI  it  can  barely  be  seen 
as  separate  from  P0.  The  two  images  will  overlap  if  the  angle  subtended 
by  the  objects  at  the  lens  is  less  than  a  =PoPi/D  =  \/AB.  This  is  called  the 
angle  of  minimum  resolution. 

Such  conditions  apply  only  to  small  sources  or  objects.  If  the  source 
is  large  each  point  will  have  a  diffraction  maximum  at  the  focus.  These 
maxima  will  overlap  and  blot  out  diffraction  effects  except  at  the  boundaries 
of  the  image. 

Stars  are  practically  point  sources  of  light.  Their  images  when  formed  by 
a  telescope  with  a  small  objective  appear  much  larger  than  when  formed 
with  a  large  telescope,  the  diameters  of  the  central  maxima  being  inversely 
proportional  to  the  diameter  AB  of  the  lens.  The  image  of  a  double  star 
formed  by  a  small  telescope  may  be  one  large  blur,  while  that  formed  by  a 
large  telescope  consists  of  two  distinct  points  of  light.  The  ability  to  sepa- 
rate the  images  of  two  small  adjacent  sources  is  called  resolving  power,  and 
may  be  shown  to  be  directly  proportional  to  the  diameter  of  the  lens,  mirror, 
or  prism  forming  the  image  —  or  to  the  cross-section  of  the  effective  beam  of 
light,  if  it  does  not  cover  the  above. 


616 


LIGHT 


If  we  bad  larger  eyes  we  could  see  much  finer  details  than  we  now  do. 
On  the  other  hand,  if  we  look  through  a  small  pin-hole  at  a  distant  light 
it  will  appear  much  larger  than  when  viewed  with  the  unaided  eye.  For 
a  similar  reason  it  is  physically  impossible  for  small  insects  to  see  details 
clearly.  To  them  an  incandescent  lamp  filament  must  appear  as  it  does  to 
us  when  we  look  at  it  through  a  very  small  pin-hole. 

707.  Diffraction  Grating. — If  there  are  a  number  of  narrow  and 
equidistant  parallel  openings  in  a  screen,  each  pair  of  openings 
will  produce  effects  similar  to  those  observed  in  Young's  double 
slit  experiment.  If  a  lens  is  placed  in  front  of  such  a  diffraction 
grating,  as  it  is  called  (Fig.  560),  the  same  path  difference  will 


Fio.  560. 

exist  between  any  pair  of  adjacent  parallel  rays.  If  a  =  AB  is  the 
distance  between  openings  and  if  the  angle  between  OP0  and  OPl 
is  6,  the  difference  of  path  between  corresponding  parts  of 
adjacent  slits  (e.g.  A  and  B)  is  AC  =  a  sin  6,  and  the  condition 
that  there  shall  be  a  maximum  at  P1  for  the  wave  length  ^  is 

a  sin  6  —  nX 

If  light  of  one  wave-length  is  used,  there  is  a  series  of  maxima 
on  either  side  of  the  axis  in  positions  where  a  sin  6  equals  1, 
2,  3,  etc.,  wave-lengths. 

If  white  light  is  used,  corresponding  maxima  for  two  different 
colors  are  at  different  distances  from  the  axis.  The  central 
maximum  P0  is  white,  as  the  condition  for  reinforcement  at 
that  point  (n=0)  is  the  same  for  all  colors.  The  other  maxima 
are  drawn  out  into  spectra  on  each  side  of  the  axis,  as  0  varies 


OPTICAL  INSTRUMENTS  AND  MEASUREMENTS          617 

with  the  wave-length.  The  value  of  the  ordinal  number  n 
determines  the  order  of  the  spectrum. 

If  6  is  a  small  angle  the  distances  between  points  in  the 
spectra  are  nearly  proportional  to  the  differences  of  the  corre- 
sponding wave-lengths,  so  that  the  spectra  formed  by  gratings 
are  said  to  be  normal,  as  contrasted  to  those  due  to  prisms,  in 
which  there  is  no  simple  law  of  distribution.  All  gratings  give 
spectra  which  are  alike  in  their  distribution  of  colors,  although 
they  may  differ  in  length.  The  lengths  of  the  spectra  in- 
crease directly  as  the  order  of  the  spectrum,  so  that  those  beyond 
the  first  overlap,  and  they  also  rapidly  diminish  in  intensity. 

A  grating  such  as  has  been  considered  above  is  called  a 
transmission  grating  and  consists  of  lines  ruled  (several  thousand 
to  the  inch)  on  glass  by  a  diamond  point.  Reflection  gratings  are 
made  by  ruling  lines  on  a  polished  surface  of  speculum  metal; 
the  incident  light  is  reflected  by  the  polished  strips  between  the 
rulings.  Such  gratings  are  ruled  by  entirely  automatic  machines. 

The  effect  due  to  a  grating  is  precisely  the  same,  so  far  as 
position  of  maxima  is  concerned,  as  that  due  to  two  slits  with  the 
same  interval  between  them.  The  intensities  of  the  grating 
spectra,  however,  are  far  greater,  the  amplitude  of  vibration 
being  in  proportion  to  the  number  of  openings. 

The  resolving  power  of  a  grating  is  also  greater.  The  width  of  a  maxi- 
mum in  the  interference  bands  given  by  two  slits  is  (§648)  w  =  D\/a.  The 
width  of  the  maxima  given  by  a  grating  having  N  openings  is  w  =  D\/Na, 
since  N  is  the  aperture  of  the  grating,  so  that  this  width  is  inversely  as 
the  breadth  of  the  grating  (§648).  Diffraction  gratings  are  generally  used 
for  measurements  of  wave-length. 

A  grating  with  crossed  lines  gives  a  beautiful  series  of  crossed  spectra. 
This  effect  may  be  observed  by  looking  through  a  handkerchief  or  umbrella 
top  at  a  distant  light.  Brilliant  diffraction  effects  are  also  obtained  by 
looking  at  a  source  through  a  cobweb  or  feather,  or  from  the  light  reflected 
from  mother  of  pearl.  In  the  latter  case  the  effect  is  due  to  striations,  as 
may  be  proved  by  transferring  the  effects  to  wax  by  pressure. 

OPTICAL  INSTRUMENTS  AND  MEASUREMENTS 

708.  The  Eye  is  an  essential  part  of  any  optical  combination. 
Like  a  photographic  camera,  it  is  a  closed  chamber  into  which 
light  can  enter  only  through  the  lens.  As  the  camera  lens  throw* 


618  LIGHT 

an  image  on  a  sensitive  photographic  plate  which  excites  the 
silver  grains,  the  lens  of  the  eye  forms  a  picture  on  the  mat  of 
sensitive  nerve  endings  covering  the  retina.  The  amount  of  light 
entering  the  camera  is  regulated  by  an  "iris"  diaphragm  of  adjust- 
able size;  similarly  the  amount  of  light  entering  the  eye  is  con- 
trolled by  the  size  of  the  pupil,  which  automatically  changes  in 
diameter  between  the  limits  of  about  2  and  5  mm.  The  parts 
of  the  eye  are  shown  in  Fig.  561.  S  is  the  sclerotic  membrane, 

the  outer  enclosure  of  the  eye.  C  is 
the  cornea,  a  strong  transparent  mem- 
brane. I  is  the  iris,  the  colored  part 
of  the  eye,  with  a  central  orifice,  the 
pupil,  which  admits  light  through  the 
crystalline  lens  L,  which  focuses 
images  on  the  retina  R.  The  nerve 
endings  covering  the  retina  run  to- 
gether like  the  strands  of  a  cable  into 
the  optic  nerve  0,  which  conveys 
stimuli  to  the  brain.  Muscles  at- 
tached to  the  periphery  of  the  lens  can  by  their  contraction 
or  relaxation  so  change  its  curvature  as  to  enable  it  to 
focus  either  distant  or  very  near  objects  on  the  retina.  This 
process  is  called  accommodation.  Two  objects  are  clearly  seen 
separately  when  the  angle  between  them  at  the  eye  is  a  little  less 
than  I/  or  the  distance  between  the  retinal  images  is  0.005  mm. 
Details  are,  therefore,  more  clearly  seen  as  an  object  is  brought 
nearer,  as  the  angle  subtended  by  it  and  the  size  of  the  retinal 
image  are  then  larger;  but  there  is  a  limit  to  the  power  of  accom- 
modation of  the  eye,  so  that  usually  no  object  nearer  than  about 
25  cm.  can  be  clearly  seen.  This  is  called  the  distance  of  most 
distinct  vision.  The  normal  adjustment  of  the  eye  when  at 
rest,  is  for  "infinity,"  as  may  be  verified  by  suddenly  raising 
the  eyes  when  they  have  been  unemployed  and  looking  toward 
distant  objects.  They  will  be  in  focus. 

Between  the  cornea  and  the  crystalline  lens  is  the  aqueous 
humor,  A,  and  between  the  lens  and  the  retina  is  the  vitreous 
humor  7,  both  transparent  fluids  with  a  mean  index  of  refraction 
equal  to  1.336.  The  lens  is  built  up  of  transparent  horny  layers, 
increasing  in  density,  hardness,  and  refractive  power  toward  the 


OPTICAL  INSTRUMENTS  AND  MEASUREMENTS         619 

center.  The  index  of  refraction  of  the  outer  layer  is  1.405 ;  of  the 
next,  1.429,  and  of  the  central  region  1.454.  The  average  index 
of  refraction  is  about  1.437.  This  increase  in  density  toward  the 
axis  serves  partly  to  correct  spherical  aberration,  which  is  also 
diminished  by  the  iris  diaphragm. 

Objects  such  as  printed  letters  can  be  clearly  seen  through  a  pin-hole  in 
a  card,  even  if  they  are  as  close  as  2  cm.  to  the  eye.  This  has  been  at- 
tributed to  an  over-correction  of  the  lens  for  spherical  aberration,  so  that 
a  narrow  pencil  passing  through  the  axis  of  the  lens  has  a  very  short  focus. 
It  is  obvious  that  so  much  over-correction  would  be  worse  than  no  correc- 
tion at  all.  As  a  matter  of  fact,  a  pin-hole  image  is  formed  on  the  retina, 
the  lens  merely  sharpening  the  effect.  The  fact  that  the  apparent  size  of 
the  object  varies  as  the  card  is  moved  back  and  forth,  the  object  remaining 
at  rest,  shows  that  the  image  is  due  mainly  to  the  pin-hole. 

709.  Vision. — The  retina  is  covered,  except  over  the  optic  nerve, 
by  a  large  number  of  very  small  fibrous  bodies,  "rods"  and 
"cones/'  nerve  endings  which  are  in  some  way  stimulated  by 
light  waves.  Over  the  optic  nerve  is  the  "blind  spot,"  so  called 
because  if  the  image  falls  on  this  part  of  the  retina  it  ceases  to 
be  visible.  By  closing  one  eye  and  looking  steadily  with  the 
other  at  one  of  two  small  objects  about  two  inches  apart,  a  distance 
may  be  found  at  which  the  other  object  will  disappear.  Excita- 
tion of  the  optic  nerve  lasts  about  one-tenth  of  a  second  after  the 
stimulus  ceases,  so  that  if  intermittent  stimuli  are  applied  at 
intervals  less  than  this'  a  steady  effect  is  produced.  This  is 
called  persistence  of  vision.  The  trail  of  the  lighted  end  of  a 
cigar  if  it  be  rapidly  moved  and  the  apparent  continuity  of  mov- 
ing pictures  depend  on  this  effect. 

Sometimes  the  normal  spheroidal  shape  of  the  lens  is  altered  so 
that  the  curvatures  are  not  the  same  in  different  planes.  Light 
from  a  point  will  then  pass  through  the  eye  as  an  astigmatic  pen- 
cil with  two  focal  lines  instead  of  a  point  image  (§685) .  Horizon- 
tal and  vertical  lines  at  the  same  distance  cannot  be  simulta- 
neously brought  into  focus.  Such  eyes  are  said  to  be  astigmatic. 
Other  defects  arise  from  change  of  curvature  or  from  loss  of  the 
power  of  accommodation.  If  eyes  are  short  sighted,  the  prin- 
cipal focus  falls  short  of  the  retina,  and  distant  objects  cannot 
be  clearly  seen.  If  they  are  long  sighted,  the  principal  focus  is  on 
or  near  the  retina,  and  images  of  near  objects  cannot  be  formed 


620  LIGHT 

on  the  retina.     For  the  first  defect  concave  spectacles  are  the 
remedy;  for  the  second  they  must  be  convex. 

In  normal  eyes  the  nerve  endings  on  which  fall  corresponding  points  of 
the  two  retinal  images  lead  to  the  same  nerve  centers,  so  that  the  two 
pictures  are  exactly  superimposed  and  a  more  intense  effect  secured  than 
with  one  eye  alone.  If  one  eye-ball  be  forcibly  twisted  out  of  position 
double  images  will  be  seen.  A  further  advantage  given  by  two  eyes  is 
that  an  object  is  viewed  from  two  slightly  different  directions,  which 
gives  the  impression  of  relief.  This  principle  is  applied  to  the  stereoscope, 
in  which  two  photographs  taken  from  slightly  different  points  of  view 
are  viewed  by  each  eye  separately.  The  two  images  will  be  superimposed 
in  such  a  manner  that  the  object  appears  to  stand  out  in  space. 

With  two  eyes  it  is  also  easier  to  estimate  distances  than  with  one. 
There  is  an  angle  between  the  two  lines  of  sight  to  the  object,  which  the 
brain  unconsciously  estimates.  In  general  the  sizes  of  objects  are  inferred 
from  their  angular  magnitudes  and  estimates  of  their  distance  based  on 
experience,  or  by  comparison  with  adjacent  objects,  such  as  trees  and 
houses,  the  sizes  of  which  are  approximately  known.  Such  estimates  are 
influenced  by  the  clearness  with  which  details  are  seen.  In  places  where 
the  atmosphere  is  unusually  clear,  as  in  Arizona,  this  leads  to  the  under- 
estimation of  distance.  Conversely,  objects  seen  in  a  fog  appear  to  be  more 
distant  than  they  are,  owing  to  the  indistinctness  of  their  details.  The  angle 
subtended  by  them,  however,  corresponds  to  the  actual  distance,  hence 
they  loom  larger  than  they  are. 

710.  Irradiation. — This  is  the  apparent  increase  in  size  of  objects  as  they 
become  brighter.    The  crescent  of  the  new  moon,   for  example,   looks 
larger  than  the  remainder  of  the  disk,  the  "  old  moon,"  which  is  illuminated 
by  the  earth  alone.     The  filament  of  an  incandescent  lamp  appears  to 
increase  in  size  as  it  passes  from  ordinary  temperatures  through  red  and 
white  heat.    This  effect  was  long  supposed  to  be  due  to  the  spreading  of 
the  retinal  image  on  account  of  stimulation  of  nerves  outside  of  its  boundaries, 
in  much  the  same  way  that  an  overexposed  photographic  image  is  affected. 
It  is  now  believed  by  some  that  the  effect  is  due  merely  to  spherical  aber- 
ration of  the  eye,  which  becomes  more  noticeable  as  the  intensity  of  the 
source  increases. 

711.  The  Simple  Microscope  or  magnifying  glass  is  a  single 
convex  lens  through  which  objects  at  or  within  the  principal 
focus  of  the  lens  are  viewed.     As  .shown  in  Fig.  562,  an  enlarged 
virtual  image  A'B'  is  formed  subtending  at  the  lens  the  same 
angle  a  as  the  object  A B.     The  linear  size  of  this  image  is  deter- 
mined from  the  relation 


OPTICAL  INSTRUMENTS  AND  MEASUREMENTS 


621 


As  the  normal  adjustment  of  the  eye  is  for  infinity,  the  object 
is  usually  at  or  very  ilear  the  principal  focus.  In  no  case  can 
the  image  be  clearly  seen  when  nearer  than  the  limit  of  distinct 
vision.  The  actual  linear  magnitude  of  the  image  counts  for 
little;  the  size  of  the  retinal  image  depends  on  the  angle  sub- 
tended at  the  eye,  and  if  the  latter  is  very  near  the  lens  this 
angle  is  substantially  that  subtended  from  the  lens.  The  lens 
simply  increases  the  power  of  accommodation  of  the  eye,  so  that 
the  object  may  be  brought  nearer  and  thus  subtend  a  greater 


Fio.  562. 

angle.  With  the  unaided  eye,  the  greatest  detail  is  observed 
at  the  distance  of  most  distinct  vision  (25  cm.)  where  it  sub- 
tends the  angle  fi  (Fig.  552.)  With  the  lens,.  the  object  is  brought 
nearer,  approximately  to  the  principal  focus,  and  the  angle 
subtended  by  it  increases  from  /?  to  a.  The  magnification 
M  of  the  retinal  image  is,  therefore,  a//?.  If  /  is  the  focal 
length  of  the  lens,  d  the  limit  of  distinct  vision, 


/?/2=2/tan  a/2 
Therefore,  if  these  angles  are  small 


712.  Power  of  Lenses.  —  The  magnifying  power  of  a  lens  is, 
as  shown  above,  inversely  proportional  to  the  principal  focal 
length,  hence  I//  is  a  measure  of  its  power.  The  practical  unit 
of  lens  power  is  that  of  a  lens  with  a  focal  length  of  one  meter. 
This  unit  is  called  a  diopter  or  dioptric.  The  power  of  converg- 
ing lenses  is  considered  positive,  that  of  diverging  lenses  negative. 
The  relation  deduced  in  §686  shows  that  the  power  of  a  number 
of  lenses  in  contact  is  the  algebraic  sum  of  their  individual 
powers. 


622 


LIGHT 


Fia.  563. 


713.  Eye-pieces. — The  part  next  the  eye  of  an  optical  train  of 
lenses,  such  as  those  of  telescopes  and  compound  microscopes, 
usually  consists  of  some  form  of  simple  microscope  known  as 
an  eye-piece.     With  a  single  lens,  much  of  the  light  from  the 
real  image  formed  by  the  objective  0,  which  is  usually  viewed 

through  the  eye-piece,  would  be 
lost.  In  order  to  avoid  this,  light 
is  gathered  in  toward  the  axis  by  a 
second  lens,  called  the  field  lens  F 
(Fig.  563).  Nearly  all  the  light 
would  pass  by  the  edge  of  the  eye 
lens  E  if  F  were  absent.  It  may 
be  shown  that  a  combination  of  two  lenses  of  the  same  kind  of 
glass  is  nearly  achromatic  if  they  are  placed  at  a  distance  from 
each  other  d=(fv  +/2)/2.  This  property  is  utilized  in  most  eye- 
pieces which  consist  of  a  field  lens  and  eye  lens. 

In  Huyghens'  eye-piece  /i  =  3/2  (Fig.  563).  Hence  d-2/a  and  if  the 
image  due  to  the  objective  and  the  first  lens  is  formed  half  way  between 
the  lenses  the  emergent  light  will  be  parallel  and  a  virtual  image  formed 
at  infinity.  If  a  cross  thread  is  used,  it  must  be  placed  at  AB.  The  lenses 
are  convex  toward  the  incident  light  and  of  such  curvature  as  to  reduce 
the  spherical  aberration  to  the  minimum. 

In  the  Ramsden  eye-piece  (Fig.  564)  /t  =/2.  If  the  lenses  are  placed 
apart  at  the  distance  (/i+/2)/2  dust  particles  on  the  field  lens  would  be 
visible  through  the  second.  In 
order  to  avoid  this,  the  lenses  are 
usually  placed  at  a  distance  of 
2//3.  The  principal  focal  point 
of  the  combination  is  at  a  distance 
//4  in  front  of  the  first  lens.  The 
object,  or  the  real  image  due  to  the 
objective,  is  at  this  point,  and  the 
final  virtual  image  is  at  infinity.  The  chromatic  aberration  is  small,  and 
the  spherical  aberration  is  reduced  by  using  plano-convex  lenses  with  con- 
vex surfaces  facing  each  other. 

In  all  these  eye-pieces  the  emergent  red  and  violet  rays  are  nearly  parallel, 
hence  the  virtual  images  formed  by  the  different  colors  subtend  very 
nearly  the  same  angle  at  the  eye,  and  are,  therefore,  of  the  same  size,  but 
not  quite  equally  sharply  focused  on  the  retina. 

714.  Compound  Microscope. — In  order  to  extend  the  limit  of 
magnification  beyond  the  point  obtainable  with  a  simple  micro- 
scope, a  combination  of  lenses  is  used.     An  enlarged  real  image 


Fia.  564. 


OPTICAL  INSTRUMENTS  AND  MEASUREMENTS 


623 


A'B'  (Fig.  555)  is  formed  by  an  object  lens  or  train  of  lenses,  and 
this  image  is  further  enlarged  by  an  eye-piece,  such  as  that  of 
Huyghens,  used  as  a  simple  microscope,  which  gives  a  virtual 
image  A"B".  The  front  lens  of  the  objective  train  is  usually 
of  the  hemispherical  form  described  in  §683,  which  has  a  great 
angular  aperture,  with  very  little  spherical  aberration.  There 
are  in  addition  a  number  of  other  lenses  of  different  shapes  and 
kinds  of  glass,  so  combined  as  to  reduce  spherical  and  chromatic 
aberration  to  a  minimum  and  to  give  a  plane  focal  surface.  A 
typical  combination  is  shown  in  Fig.  566. 


Fio.  565.  Fio.  566. 

The  magnifying  power  of  the  objective,  of  focal  length  flt  is 


That  of  the  eye-piece  is,  as  shown  in  §711, 
Af,-/,//,-*//, 

where  d  is  the  minimum  distance  of  distinct  vision.     The  magni- 
fication due  to  the  combination  is 

M  =  M1Mj  =  /3/0  =  Z/d//1/2  approximately 
where  L  is  the  distance  between  the  objective  and  the  eye-piece. 

The  minimum  distance  between  two  small  objects  A  and  B  seen  through 
a  microscope  which  will  permit  of  clear  separation  of  their  diffraction 
images  is  obtained  by  a  slight  modification  of  the  expression  found  for  the 
minimum  angle  of  resolution,  a=\/AB  (§706).  The  minimum  value  which 
d  can  have  is  thus  found  to  be  X/2,  when  the  object  is  at  the  surface  of 
the  lens.  Since  this  distance  is  proportional  to  the  wave-length,  details 
which  may  be  clearly  seen  when  the  object  is  illuminated  by  blue  light  will 
be  indistinct  when  red  light  is  used. 


624  LIGHT 

715.  Astronomical  Telescope. — The  object  glass  of  a  telescope 
forms  a  real  and  of  course  greatly  reduced  image  A'B'  of  a  dis- 
tant object  (Fig.  567).  The  object  and  its  image  subtend  the 
same  angle  a  at  the  objective,  and  the  object  subtends  practic- 
ally the  same  angle  a  if  viewed  directly  by  the  eye.  If,  however, 
the  eye  views  the  image  formed  by  the  objective  at  the  distance 
of  most  distinct  vision,  (from  the  point  E),  this  image  will 
subtend  an  angle  /?  which  is  larger  than  a,  and  the  apparent 
magnification  is  Ml  =  ^/a.  When  this  image  is  viewed  through 


an  eye-piece,  (the  eye  now  being  at  E2)  ,  there  is  further  magnifi 
cation,  the  image  subtending  the  larger  angle  7-.  The  magnifi 
cation  due  to  the  combination  is 


The  limiting  angle  of  resolution  between  two  linear  sources  is  proportional 
to  X/A,  where  A  is  the  diameter  of  the  objective  (§706). 

For  astronomical  purposes  there  is  no  disadvantage  arising 
from  the  fact  that  an  inverted  image  is  formed  by  a  telescope, 
but  when  the  instrument  is  to  be  used  for  terrestrial  purposes 
it  is  necessary  to  add  an  additional  lens  or  pair  of  lenses  to 
rein  vert  the  real  image  formed  by  the  objective.  This  adds 
inconveniently  to  the  length  of  the  tube.  If  the  image  is  inverted 
by  reflection  from  a  combination  of  prisms  the  length  may  be 
diminished,  but  for  most  purposes  where  only  small  magnifica- 
tion is  required  the  form  of  telescope  devised  by  Galileo  is  most 
convenient. 

716.  Dutch  or  Galilean  Telescope.  —  This  type  is  used  for 
opera  glasses  and  for  marine  glasses.  As  shown  in  Fig.  568  an 


OPTICAL  INSTRUMENTS  AND  MEASUREMENTS 


625 


erect  virtual  image  is  formed,  the  magnification  being  M—fJf2 
(§715).  The  tube  has  a  length  approximately  equal  to  the 
difference  between  the  focal  lengths  of  the  objective  and  the 
eye-piece,  while  in  the  ordinary  telescope  the  length  is  the  sum 
of  these  distances. 


Fio.  568. 


717.  Reflecting  Telescope. — The  objective  lens  may  be  re- 
placed  by   a  large   concave   mirror.     In   this  way  chromatic 
aberration  may  be  entirely  avoided,  but  spherical  aberration  is 
more  troublesome  than  with  refractors.     As  the  real  image  is 
formed  along  the  axis  of  the  mirror  and  in  the  path  of  the  incident 
light,  special  devices  are  necessary  in  order  to  view  it.     In  the 
Newtonian  telescope  the  image  is  reflected  to  one  side  by  a  small 
right-angled  prism  which  cuts  off  very  little  light,  and  is  viewed 
by  an  eye-piece  in  the  side  of  the  tube.     Herschel  tipped  the 
mirror  slightly  so  that  the  image  was  formed  at  the  edge  of  the 
open  end  of  the  tube,  at  which  point  the  eye-piece  was  fixed.     In 
other  forms  a  small  mirror  in  the  axis  reflects  the  image  back 
into  an  eye-piece  set  in  the  center  of  the  objective  itself,  so  that  it 
can  be  viewed  from  behind. 

718.  Photographic  Camera. — This  is  a  form  of  camera  obscura 
in  which  the  image  formed  by  a  lens  falls  on  a  sensitive  photo- 
graphic plate.     The  requirements  demanded  for  the  lens  are  exact- 
ing and  in  some  cases  contradictory  to  each  other.     It  must  give 
images  free  from  spherical  and  chromatic  aberration,  and  in  many 
cases  have  great  light  power  and  a  large  field  of  view.     The  focal 
surface  must  be  plane,  and  the  magnification  must  be  the  same 
in  all  parts  of  this  plane,  so  that  no  distortion  is  produced.     The 
depth  of  focus  must  be  great,  that  is,  objects  at  different  dis- 

40 


626 


LIGHT 


tances  must  have  images  approximately  in  focus  at  the  same  time 
on  the  plate.  As  the  film  is  most  sensitive  for  the  shortest 
waves,  the  lenses  must  be  corrected  for  the  violet  and  the  yellow, 
instead  of  blue  and  red.  A  diaphragm  with  a  small  opening  is 
used  in  front  of  the  lens,  if  it  is  a  single  achromatic  combination, 
such  as  is  used  for  landscape  work.  This  reduces  spherical  aberra- 
tion and  at  the  same  time  gives  a  greater  depth  of  focus  (ap- 
proximating to  the  principle  of  the  pin-hole  camera,  in  which 
the  focus  is  nearly  independent  of  the  distance).  A  diaphragm 


Fio.   569. 


Fio.  570. 


with  a  single  lens  results  in  a  distorted  image,  however,  as  shown 
in  Fig.  569,  which  represents  the  distortion  of  the  image  of  a 
quadrilateral  network  with  the  diaphragm  in  front  of  the  lens, 
and  Fig.  570,  which  gives  the  effect  due  to  a  diaphragm  behind 
the  lens.  The  cause  is  readily  seen  to  be  due  to  the  difference 
in  deviation  of  pencils  passing  through  the  center  and  the  edge 
of  the  lens  respectively.  If  two  lenses  are  used,  with  the  dia- 
phragm] at  the  optical  center  of  the  combination,  these  distor- 
tions correct  each  other. 


Fia.  571. 

As  shown  in  Fig.  571,  the  perfect  symmetry  of  the  incident  and  the  trans- 
mitted secondary  axes  AA',  BB',  CC',  etc.,  with  respect  to  the  opening  0 
shows  that  the  distances  AB,  BC,  etc.,  in  the  object  are  in  the  same  ratio 
as  the  corresponding  distances  A 'B',  B'C',  etc.,  in  the  image,  so  that  there 
is  no  distortion,  and  if  A,  B,  C  are  in  the  same  plane,  A'B'C',  etc.,  must  be 
in  the  same  plane.  Such  lenses  are  called  rectilinear  or  orthoscopic  doublets. 

The  size  of  the  photographic  image  of  a  distant  object  is  nearly  proper- 


OPTICAL  INSTRUMENTS  AND  MEASUREMENTS 


627 


tional  to  the  focal  length  of  the  lens.  It  is,  however,  inconvenient  to  give 
a  great  length  to  the  camera  box.  This  difficulty  is  avoided  by  the  use  of 
the  teleobjective,  in  which  a  concave  lens  L2  is  placed  behind  the  converging 
lens  L  (Fig.  572.)  The  divergent  effect  of  this  lens  gives  a  virtual  focal 
length  equal  to  PF,  while  the  camera  box  has  the  much  smaller  length  LF. 
A  greatly  enlarged  image  is  secured,  but  the  field  of  view  is  reduced. 


FIG.  572. 

719.  The  Projection  Lantern  is  used  to  throw  an  enlarged  image  A'B' 
of  more  or  less  transparent  objects  on  a  screen.  The  object  AB  (Fig. 
573)  is  also  illuminated  by  a  condenser  C,  consisting  of  two  thick  plano- 
convex lenses,  with  convex  sides  facing  each  other.  The  sources  are  usually 
the  electric  arc  or  the  calcium  light.  The  focusing  lens  L  is  generally  of  the 


photographic  doublet  type,  hi  order  that  an  undistorted  image  may  be 
formed  on  the  screen.  The  object  of  the  condenser  is  not  only  to  illumi- 
nate the  object,  but  also  to  enlarge  the  field  beyond  the  limit  which  would 
otherwise  be  set  by  the  cross-section  of  the  focusing  lens. 

720.  The  Spectroscope  (Fig.  574)  is  an  instrument  for  analyz- 
ing complex  radiations  by  prismatic  dispersion  (§672)  or  by  the 
diffraction  grating.  In  order  to  secure  as  complete  separation  of 
the  colors  as  possible,  or  a  "pure"  spectrum,  a  narrow  slit  must  be 
used  as  a  source,  so  that  the  colored  images  of  the  slit  will  overlap 
as  little  as  possible.  The  resolving  power  must  be  so  great  that 
the  diffraction  images  of  the  slit  or  "lines"  do  not  overlap,  and 
this  requires  large  apertures  for  the  lenses  and  prism  or  grating 
(§707) .  The  larger  the  dispersion  the  more  complete  the  separa- 
tion of  the  image's.  For  given  dispersion,  the  length  of  the  spec- 
trum is  proportional  to  the  focal  length  of  the  observing  telescope, 
but  this  merely  affects  the  scale  of  the  spectrum,  not  the  resolu- 
tion of  the  lines  or  the  clearness  of  detail. 


628  LIGHT 

The  plan  of  the  ordinary  form  of  spectroscope  is  shown  in  Fig. 
574.  The  essential  parts  are:  A  narrow  slit,  S;  a  collimating  lens 
C,  which  converts  the  wedge  of  light  from  the  slit  into  a  parallel 
beam;  a  prism  to  disperse  the  colors;  a  telescope  lens  T,  with 
which  real  images  of  the  slit  are  produced  at  the  focus  of  the 
eye-piece  E.  If  light  of  an  infinite  number  of  colors  is  emitted  by 
the  source,  the  infinite  number  of  partially  overlapping  images 
forms  a  continuous  spectrum;  if  only  a  finite  number  of  colors  are 
emitted,  there  will  be  a  finite  number  of  slit  images,  giving  a 
discontinuous  or  line  spectrum.  A  homocentric  pencil  (cone) 
incident  on  a  prism  remains  homocentric  after  transmission  at 
the  angle  of  minimum  deviation;  for  all  other  angles  of  trans- 
mission, it  becomes  astigmatic,  and  no  true  image  is  produced. 


Fio.    574. 

The  condition  of  homocentricity  cannot  be  fulfilled  for  all  colors 
simultaneously  except  when  the  incident  light  is  parallel,  in 
which  case  light  of  each  color  emerges  in  a  parallel  beam.  For 
this  reason  the  collimator  is  necessary.  This  lens  must  be  achro- 
matic. So  far  as  purity  of  spectrum  is  concerned,  it  is  evidently 
unnecessary  for  the  telescope  lens  to  be  achromatic,  but  it  is 
usually  corrected,  in  order  that  all  the  colors  may  be  at  once 
in  the  focus  of  the  eye-piece.  Positions  in  the  spectrum  may  be 
referred  to  the  image  of  a  scale  R  reflected  from  the  side  of  the 
prism. 

This  instrument  is  called  a  spectrograph  when  the  telescope  is 
replaced  by  a  camera  for  photographing  the 'spectrum,  and  a 
spectrometer  when  provided  with  a  graduated  circle  for  measuring 
the  angular  deviation  of  the  light.  The  direct-vision  spectroscope 
usually  made  in  small  sizes  for  pocket  use,  has  a  combination  of 


OPTICAL  INSTRUMENTS  AND  MEASUREMENTS 


629 


crown  and  flint  prisms,  as  shown  in  Fig.  575.  The  mean  deviation 
is  zero,  but  there  is  some  residual  dispersion  which  gives  a 
short  spectrum  (§676). 

A  plane  diffraction  grating  may  replace  the 
prism  of  a  spectroscope,  or  spectrometer. 
With  the  latter  the  angular  deviations  of  the 
diffraction  maxima  may  be  measured  and  the 
wave-lengths  determined  by  the  relation  deduced  in  §707. 

721.  The  Concave  Grating  was  Rowland's  greatest  contribution  to  spec- 
troscopy.  The  lines  are  ruled  at  equal  distances  on  the  surface  of  a  con- 
cave mirror  of  speculum  metal  which  focuses  as  well  as  diffracts  the  light 
If  R  is  the  radius  of  curvature  of  the  mirror  (Fig.  576)  and  if  the  slit  is 
at  any  point  on  the  circumference  of  a  circle  having  the  radius  as  its 
diameter,  it  is  found  that  the  spectra  of  all  orders  are  in  focus,  along  the 
circumference  of  this  same  circle,  so  that  no  lenses  are  necessary.  Usually 
the  grating  G  a  mounted  at  one  end  and  the  eye-piece,  or  camera,  E  at  the 
other  end  of  a  beam  R  equal  in  length  to  the  radius  of  the  grating.  This 
beam  has  a  swivel  truck  under  each  end,  which  travels  on  tracks  at  right 
angles  to  each  other,  with  the  slit  at  the  intersection  S.  The  distance 
SE  between  the  slit  and  the- eye-piece  are  proportional  to  a  sin  0  =  nA  and 
therefore  are  proportional  to  the  wave-lengths  of  the  part  of  the  spectrum 
in  the  field  of  view. 


FIQ.  576. 

722.  Michelson's  Interferometer. — The  surface  of  the  glass 
plate  Pl  (Fig.  577)  is  "half  silvered,"  that  is,  the  silver  film  is 
of  such  thickness  that  about  one-half  of  the  incident  light  is 
reflected.  Light  from  the  point  S  of  an  extended  source  falls 
on  this  surface  at  A  and  is  in  part  transmitted  to  the  mirror 
Mlt  in  part  reflected  to  the  mirror  M2.  From  these  mirrors  it 
will  be  reflected,  retrace  its  course,  and  some  will  finally  reach 


630  LIGHT 

the  eye  at  E.  If  Ml  and  M2  are  at  the  same  optical  distance 
from  S,  and  if  each  is  perpendicular  to  the  rays  that  fall  on  it 
the  light  will  appear  to  come  from  two  exactly  superimposed 
images  of  the  source  and  there  will  be  no  interference.  The 
plate  Pa  is  introduced  merely  to  give  the  ray  SAM^  the  same 
path  in  glass  as  the  ray  SAM  2,  so  that  the  optical  and  the  geo- 
metrical paths  will  be  the  same.  Now  if  monochromatic  light 
of  wave-length  A  be  used  and  if  M  l  be  displaced  a  distance  A/  4, 
the  waves  that  reach  the  eye  will  have  traversed  paths  that 
differ  by  XI  2  and  will  destroy  one  another  in  the  center  of  the 
field  of  view.  A  further  displacement  of  A/  4  will  restore  the 
light.  Thus  by  slowly  displacing  Ml  and  counting  the  number 
of  times,  N,  that  the  light  reaches  a  maximum,  the  distance,  d, 
through  which  Ml  has  been  displaced  may  be  found  from  the 
relation 


Michelson  has  by  this  instrument  measured  the  length  of  the 
standard  meter  of  Paris  in  terms  of  wave-lengths  of  several  of 
the  spectral  lines  of  cadmium,  with  an  accuracy  of  about  one 
part  in  ten  millions.  The  wave-lengths  of  such  lines  are  probably 
the  most  permanent  and  unchangeable  standards  of  length 
which  can  be  obtained.  The  interferometer  has  been  used  to 
make  numerous  other  measurements  of  great  accuracy.  For 
example,  the  thickness  of  the  thinnest  water  films  has  been  found 
by  means  of  it.  .If  a  film  of  thickness  d  be  introduced  in  the 
path  of  one  of  the  rays  the  change  of  optical  path  (§678)  will 
be  2(n—  l)d.  This  distance  being  measured  by  the  interferome- 
ter and  n  being  known,  d  was  deduced. 

The  above  account  is  incomplete,  since  it  was  confined  to  what  is  observed 
in  the  center  of  the  field  of  view,  the  mirrors  being  adjusted  so  that  the 
image  of  M  A  in  the  plate  Pl  is  parallel  to  M  2.  In  this  case  the  whole  effect 
is  the  same  as  when  light  is  reflected  from  a  thin  film  with  parallel  sides 
(§695).  Around  a  central  point  there  are  circular  fringes  which  change  in 
diameter  as  M  t  is  moved.  If  Afa  and  the  image  of  M  t  are  not  quite  parallel 
the  effects  will  be  similar  to  those  due  to  a  wedge-shaped  film  (§695).  The 
fringes  Will  still  be  approximately  circular  but  not  concentric,  the  centers 
being  on  a  line  perpendicular  to  the  edge  of  the  wedge.  As  M  l  moves 
the  fringes  will  sweep  past  any  point  in  the  field  of  view  and  it  is  this  suc- 
cession of  fringes  that  is  usually  counted  in  the  use  of  the  instrument. 


EMISSION  OF  RADIANT  ENERGY 


631 


S 


The  interferometer  is  also  used  to  measure  the  difference  of  wave-length 
of  two  spectral  lines  which  are  so  close  together  that  they  cannot  be  separated 
by  a  grating.  Light  from  such  a  composite  source  produces  two  series  of 
nearly  coincident  fringes.  These,  as  Ml  is  being  displaced,  coincide  at 
equal  intervals  depending  on  the  difference  of  wave-length  and  then 
produce  maxima  of  visibility.  The  red  lines  of  cadmium  are  apparently 
single  lines. 

723.  Lummer-Brodhun  Photometer. — A  cube  C  is  made  of  two  right- 
angled  prisms,  as  shown  in  -Fig.  578.     The  hypothenuse  surface  of  the  one 
prism  is  plane,  that  of  the  other  convex,  with  the  vertex  ground  flat.     These 
two  surfaces  are  in  close  contact.    The  sources  to  be  compared,  Si  and  £2, 
are  mounted  on  an  optical  bench,  over  the  center  of  which  is  the  white  screen 
W  of  paper  or  gypsum.     The  diffuse  illumination  from  this  screen  is  re- 
flected from  the  mirrors  M i  and  M2  through  the  prism  faces  AB  and  CD. 
Light  from  Si  and  Sz  is  transmitted  with- 
out loss  through  the  area  of  contact  of 

the  two  prisms,  and  is  totally  reflected 
from  the  air  film  between  those  parts  of 
the  hypothenuse  surfaces  which  are  not 
in  contact.  If  a  telescope  is  focused  on 
the  region  of  contact  through  the  side 
CD,  light  will  enter  it  from  Si  by  trans- 
mission and  from  S2  by  reflection.  The 
field  will  be  uniformly  illuminated  if  the 
two  sides  of  the  screen  W  are  equally 
illuminated;  otherwise  the  area  of  con- 
tact will  appear  brighter  or  darker  than 
the  surrounding  part  of  the  field. 

724.  Standards    of    Luminosity. — For 
accurate    and    easily  reproducible  com- 
parisons  of  photometric   measurements 

constant  and  easily  attainable  standards  are  necessary.  So  far  no  abso- 
lutely reliable  standard  source  has  been  found.  For  ordinary  purposes  the 
British  standard  candle  is  used.  These  are  made  of  sperm,  weigh  six  to  the 
pound,  and  are  normally  supposed  to  burn  120  grains  per  hour.  In  actual 
practice  there  are  great  deviations  from  uniformity. 

Other  standards  in  use  are  the  Methven  screen  and  the  Hefner-Alteneck 
lamp.  The  former  is  a  gas  flame  from  an  Argand  burner,  the  light  from 
which  passes  through  a  rectangular  opening  of  definite  size.  The  latter  is 
the  flame  of  a  lamp  burning  amyl  acetate.  In  both,  the  flame  should  be 
kept  at  a  definite  and  constant  height.  The  light  from  these  sources  is 
constant  within  a  few  per  cent. 

EMISSION  OF  RADIANT  ENERGY. 

725.  Analysis  of  Radiation. — The  methods  by  which  radiation 
may  be  analyzed  by  the  dispersion  of  colors,  or  waves  of  different 


FIG.  578. 


632  LIGHT 

length,  have  already  been  described  (§720).  The  radiation  from 
all  known  sources  is  complex — that  is  to  say,  it  contains  waves 
of  more  than  one  frequency  of  vibration.  The  principal  types 
of  emission  spectra  may  be  observed  side  by  side  if  the  image  of 
a  long  electric  arc  is  focused  on  a  slit  beyond  which  a  prism  and 
lens  are  placed  so  that  a  large  spectrum  is  thrown  on  the  screen. 
The  light  coming  from  the  positive  carbon  forms  a  brilliant 
continuous  spectrum  including  all  the  colors.  Next  to  it  is  the 
discontinuous  spectrum  of  the  arc  proper,  the  luminous  flame, 
which  contains  the  vapors  of  carbon,  various  compounds  of 
carbon,  and  any  metals  that  may  be  present  as  impurities  in  the 
electrodes.  This  spectrum  consists  of  a  number  of  narrow  lines 
due  to  the  metals  present,  and  several  groups  of  bands,  each 
composed  of  a  large  number  of  fine  lines  so  spaced  as  to  produce 
the  effect  of  the  shading  of  fluted  columns  in  a  line  drawing — 
hence  they  are  often  referred  to  as  fluted  bands  (see  Fig.  582). 
These  appear  to  be  due  to  the  vapors  of  carbon  or  the  com- 
pounds of  carbon.  The  bands  are  especially  strong  in  the 
violet,  and  this  gives  the  arc  its  characteristic  violet  color.  The 
violet  bands  appear  to  be  mainly  due  to  cyanogen  or  some  other 
compound  of  carbon  with  nitrogen,  as  they  are  very  weak  if  the 
arc  is  deprived  of  nitrogen.  Next  to  this  is  the  spectrum  of  the 
negative  electrode,  which  is  at  a  much  lower  temperature  than 
the  positive,  in  which  there  is  very  little  blue  or  violet. 

As  illustrated  by  the  arc  spectrum,  there  are  two  general  types 
of  emission  spectra,  the  continuous  and  discontinuous,  and  the 
latter  in  turn  may  be  divided  into  line  and  band  spectra. 
§§731,  732. 

726.  Invisible  Radiations. — It  was  found  by  William  Herschel 
in  1800  that  if  a  sensitive  thermometer  is  placed  in  any  part  of 
the  spectrum  of  the  sun  it  will  show  a  rise  of  temperature,  this 
effect  increasing  in  going  from  violet  to  red.  It  does  not,  how- 
ever, cease  abruptly  at  the  boundary  of  the  visible  spectrum, 
but  increases  to  some  distance  beyond  it,  and  then  gradually 
diminishes,  the  observed  limit  in  any  case  depending  on  the 
sensitiveness  of  the  thermometer.  Evidently  there  is  radiation 
which  is  less  refracted  than  the  red,  and  which  by  analogy  we 
may  conclude  has  waves  of  greater  length  than  those  of  red  light. 
It  was  shown  by  Herschel  tnat  this  "radiant  heat"  is  subject  to 


EMISSION  OF  RADIANT  ENERGY  633 

the  same  laws  of  reflection  and  refraction  as  light,  but  their 
identity  was  not  generally  accepted  until  nearly  fifty  years  later, 
when  it  was  shown  that  the  invisible  radiation  is  capable  of  pro- 
ducing interference  effects,  and  that  it  is  likewise  capable  of 
dispersion  and  polarization.  As  the  ideas  in  regard  to  the  nature 
of  heat  crystallized  it  was  seen  that  heat  can  be  associated  only 
with  matter,  but  that  the  energy  of  this  heat  may  be  partly 
transformed  into  the  energy  of  ether  waves,  and  this,  if  absorbed 
by  matter,  will  again  appear  as  heat.  It  thus  becomes  clear  how 
invisible  radiations  from  a  hot  body  can  pass  through  an  ice  lens 
without  melting  it,  and  set  fire  to  a  piece  of  paper  at  the  focus. 
The  name  infra-red  is  applied  to  these  long-wave  radiations. 

The  existence  of  ultra-violet  radiations  in  the  solar  spectrum, 
with  waves  shorter  than  those  of  violet,  is  shown  by  means 
of  the  chemical  effect  produced  on  chloride  of  silver.  Photo- 
graphic films  are  very  sensitive  to  the  ultra-violet  radiation, 
which  is  especially  active  in  its  chemical  effects.  It  also  excites 
strong  fluorescence  (§750)  in  many  substances.  If  a  strip  of 
paper  moistened  in  acidulated  sulphate  of  quinine  solution  is 
held  in  the  arc  spectrum  the  excited  fluorescent  light  shows  the 
existence  of  ultra-violet  radiation  in  the  spectra  of  both  the 
positive  carbon  and  the  arc  proper. 

The  non-visibility  of  the  infra-red  and  ultra-violet  radiations 
is  due  merely  to  the  limitations  of  the  eye.  The  eye  will 
"  resonate  "  to  vibrations  between  certain  limits  of  frequency,  the 
photographic  film  or  fluorescent  screen  to  certain  others;  but  if 
the  receiving  surface  is  blackened  the  energy  of  waves  of  all 
frequencies  is  almost  completely  absorbed,  and  by  the  amount 
of  heat  developed  we  may  determine  the  amount  of  energy  in 
any  part  of  the  spectrum. 

727.  Methods  of  Detecting  Invisible  Radiation. — Photography  is  a  thor- 
oughly satisfactory  method  of  detecting  even  the  shortest  ultra-violet 
radiations  so  far  discovered.  This  method  cannot  be  used  however,  at 
the  opposite  end  of  the  spectrum,  as  it  seems  impossible  to  make  any 
photographic  film  which  is  sensitive  to  the  infra-red — it  is  difficult  to  make 
one  which  will  even  reach  the  limit  of  the  visible  red  For  this  reason 
other  less  satisfactory  methods  must  be  employed,  which  are  usually  based 
on  the  heating  effects  produced.  For  one  of  the  earliest  instruments  for 
the  detection  of  infra-red  radiation,  the  thermopile,  as  well  as  other  more 
recent  modifications,  see  §§333,  482. 


634 


LIGHT 


728.  Continuous  Spectra. — It  is  a  familiar  experience  that  as 
the  temperature  of  a  body  rises  it  first  reaches  a  dull  red  heat, 
then  yellow,  and  finally  a  dazzling  white.  Conversely,  if  the 
spectrum  of  the  positive  electrode  of  an  arc  light  is  thrown  on  a 
screen,  and  if  the  current  is  suddenly  cut  off,  it  will  be  observed 
that,  as  the  carbon  cools,  violet,  blue,  green,  and  yellow  disappear 
in  succession,  and  finally  the  red.  If  a  sensitive  thermopile  is 
placed  far  in  the  infra-red  it  will  be  found  that  sensible  radiation 
is  still  emitted  long  after  the  luminosity  has  disappeared. 


7377°  aba. 


\  =  10,000 


20,000 


30,000      40,000 
Fio.  579. 


50,000 


60,000 


Draper  (1847)  found  that  all  bodies  begin  to  glow  at  about  the 
same  temperature.  The  actual  temperature  in  any  case  depends 
somewhat  on  the  sensitiveness  of  the  eye,  but  is  not  far  from 
400°.  Draper's  law  is  approximately  true  for  all  colors  and 
temperatures — that  is  to  say,  all  solids  begin  to  radiate  red,  or 
yellow,  or  violet,  or  any  particular  "heat  color"  in  the  infra-red 
at  the  same  temperature.  The  spectral  distribution  of  energy 


EMISSION  OF  RADIANT  ENERGY 


635 


may  be  shown  by  plotting  a  curve  with  wave  lengths  as  abscissae 
and  ordinates  proportional  to  the  galvanometer  deflections 
observed  as  the  thermopile  passes  through  the  spectrum. 
Fig.  579  shows  a  series  of  such  curves  for  temperatures  ranging 
from  836°  to  1377°  absolute,  the  source  consisting  of  a  strip 
of  blackened  metal  electrically  heated.  The  depressions  in  the 
curves  are  due  to  absorption  by  carbon  dioxide  and  water  vapor. 
The  general  character  is  the  same  for  all  solids,  but  differences  in 
the  ordinates  may  arise  from  differences  in  the  state  of  the  surface, 
whether  black,  or  rough,  or  polished,  etc.  Investigation  shows 
that  there  is  a  very  definite  relation  between  the  absolute 
temperature  of  the  source  if  it  is  black,  or  approximately  so,  and 
the  wave  length  corresponding  to  the  maximum  ordinate  of  the 
energy  curve,  such  that  XmT  =  Constant  (see  §729).  This  con- 
stant varies  slightly  from  2814  at  621.2°  to  2928  at  1646°,  with  a 
mean  value  of  2879.  The  unit  of  wave  length  is  the  micron,  //, 
or  .001  mm. 

The  total  energy  emitted  by  an  incandescent  source  is  propor- 
tional to  the  area  included  between  the  energy  curve  and  the 
axis  of  X.     That  part  of  the  energy  which  produces  luminosity 
is  included  between  the  limits  of  the 
visible  spectrum.     The  luminous  effi- 
ciency  is  proportional  to  the  ratio 
between  these  two  areas.     Fig.  580 
illustrates  the  relative  luminous  effi- 
ciencies of  the  positive  pole  of  an  arc 
light,  of  an  incandescent  light,  and  of 
a  piece  of  red  hot  carbon.     The  lumin- 
ous   energy    is   represented    by  the 
shaded  part  of  the  area  between  the 

curve  and  the  ^-axis.  Evidently  the  luminous  efficiency  rises 
very  rapidly  with  the  temperature,  so  that  a  small  increase  in 
the  current  through  an  incandescent  lamp  will  greatly  increase 
its  brightness,  and  conversely.  The  luminous  efficiency  of  the 
arc  is  about  10 percent.;  of  incandescent  lamps,  3  to  5  per  cent.; 
of  gas  and  candle  flames  2  or  3  per  cent.  Luminous  vapors, 
which  radiate  "  selectively,"  are  usually  more  efficient  than  solids. 

729.  Law  of  Radiation. — It  may  be  assumed  that  ether  waves  are  set  up 
by  agitation  of  the  electrons,  or  ions  in  the  case  of  the  long  waves,  within  the 


636  LIGHT 

molecules  of  matter,  and  that  the  frequencies  of  vibration  of  these  electrons 
or  ions  is  dependent  upon  the  kinetic  energy  of  the  molecules  (§262). 
In  solids  the  molecules  are  so  close  that  there  can  be  little  chance  for  the 
electrons  to  vibrate  without  constraint  with  their  natural  periods.  Owing 
to  frequent  collisions,  a  wide  range  of  velocities  and  vibration  frequencies 
must  exist.  The  forced  ether  vibrations  will  have  periods  corresponding 
to  those  of  the  electrons.  The  greater  part  of  the  radiant  energy  will  be 
due  to  the  large  number  of  molecules  which  have  velocities  in  the  neighbor- 
hood of  the  mean  velocity.  There  will  be  relatively  few  molecules  which 
will  have  extreme  velocities,  either  large  or  small,  and  therefore  the  longest 
and  shortest  ether  waves  will  have  relatively  a  small  amount  of  energy.  As 
the  temperature  rises  there  will  be  a  general  increase  of  kinetic  energy, 
many  molecules  moving  faster  but  none  slower  than  before,  so  that  both 
the  maximum  energy  and  the  maximum  rate  of  vibration  of  the  excited 
ether  waves  will  move  toward  the  violet  end  of  the  spectrum.  The  source 
will  rise  to  red  and  finally  to  white  heat.  It  is  thus  evident  that  a  spectral 
intensity  curve  must  be  a  sort  of  probability  curve  based  on  the  distribution 
of  velocities  of  the  molecules  with  its  ordinates  exaggerated  on  the  side  of 
the  violet,  as  illustrated  in  the  curves  of  Fig.  579. 

By  such  reasoning  Wien  theoretically  deduced  the  relation  ^m7T  =  con- 
stant, known  as  Wien's  law.  Later  Planck  established  a  general  relation 
between  the  intensity  of  radiation  corresponding  to  a  given  wave-length  ^, 
and  the  absolute  temperature  of  the  source,  as  follows: 


where  e  is  the  base  of  the  natural  system  of  logarithms,  T  the  absolute 
temperature,  and  C  and  c  constants.  This  relation  holds  within  wide  limits 
for  bodies  which  are  black  or  approximately  so.  For  such  bodies  the  value 
of  c  =  1.4598,  C  =  3.7179  XlO-5.  The  law  XmT  **  constant  -c/ 5  may  be  de- 
duced by  differentiating  the  above  expression  for  a  maximum  value  of  7j, 
and  Stefan's  law  by  integrating  the  intensity  over  the  whole  spectrum  (see 
§337). 

730.  Discontinuous  Spectra  were  noted  by  a  number  of  ob- 
servers during  the  first  half  of  the  nineteenth  century,  and  it  was 
at  least  dimly  realized  that  in  some  cases  the  appearance  of  the 
spectrum  is  characteristic  of  the  substances  emitting  the  radia- 
tion. There  are  two  types  of  discontinuous  spectra,  known 
respectively  as  line  and  band  spectra. 

In  1860  Kirchhoff  and  Bunsen  established  definitely  the  law  that 
all  gases  and  vapors  give  discontinuous  spectra,  and  that  these 
spectra  are  perfectly  characteristic  of  the  substance.  They  dis- 
covered rubidium  and  caesium  by  the  application  of  this  principle, 


EMISSION  OF  RADIANT  ENERGY  637 

which  has  been  fruitful  in  the  discovery  of  other  new  elements, 
notably  helium  and  the  rare  atmospheric  gases  in  recent  times. 

At  first  only  the  spectra  of  flames  were  studied,  but  later  it  was 
found  that  the  electric  spark  between  metallic  terminals  gives 
lines  due  both  to  the  electrodes  and  to  the  surrounding  atmos- 
phere, and  that  if  the  electric  discharge  passes  through  a  gas  in  a 
partially  exhausted  tube  (vacuum  tube)  the  luminosity  is 
confined  to  the  gas,  and  the  metallic  lines  disappear.  Only  in 
exceptional  cases  is  it  possible  to  make  a  gas  luminous  except  by 
the  electric  discharge. 

It  would  seem  reasonable  to  imagine  that  an  electron  attached 
to  an  atom  has  its  own  definite  rate  or  rates  of  vibration,  just  as 
a  tuning  fork  has  one  definite  period  and  a  piano  wire  several. 
In  a  solid  or  liquid  constraints  and  collisions  may  produce  forced 
vibrations  covering  a  wide  range  of  periods,  but  in  gases  or  vapors, 
where  collisions  must  be  comparatively  few,  there  is  a  prepon- 
derance of  free  vibrations.  If  an  electron  has  one  free  period  of 
vibration,  say  that  corresponding  to  the  color  of  luminous  sodium 
vapor,  there  will  be  but  one  image  of  the  slit  if  the  D  lines  were 
single — yellow  in  the  case  assumed.  If  there  are  a  number  of 
coexisting  vibrations  of  different  periods  there  will  be  an  equal 
number  of  spectral  lines. 

731.  Line  Spectra  are  given  by  the  metals  and  salts  of  the 
sodium  and  calcium  groups  in  the  Bunsen  flame,  and  also  by  a 
number  of  other  metals  if  spray  from  solutions  of  their  salts,  or 
ions  caused  by  the  electric  spark,  are  passed  into  the  flame.  The 
spectrum  of  the  electric  spark  or  arc  between  electrodes  composed 
of  or  coated  with  any  metals  or  their  salts  contains  many  more 
lines  than  that  of  the  flame.  The  number  may  be  very  large, 
ranging  from  a  dozen  or  so  in  the  case  of  the  alkali  metals  to 
many  thousands,  in  the  cases  of  iron  and  uranium. 

It  seems  that  no  two  elements  have  any  common  lines,  but  the 
spectrum  of  a  given  element  may  show  differences  in  the  number 
and  the  appearance  of  the  lines  according  to  the  nature  of  the 
source,  whether  flame,  arc,  or  spark.  The  middle  part  of  Fig. 
581  shows  the  arc  spectrum  of  iron,  the  lowest  part  is  the  spark 
spectrum  of  the  same  metal,  the  highest  part  is  the  solar  spectrum, 
which  shows  many  absorption  lines  coinciding  with  the  emission 
lines  of  iron.  All  salts  of  the  same  metal  give  the  same  line 


638 


LIGHT 


spectrum,  although  in  some  cases  they  give  band  spectra  as  well, 
which  may  be  characteristic  of  the  salt  (see  next  section).  The 
lines  of  the  non-metallic  components  do  not  appear  with  those 
of  the  metal  except  in  rare  cases.  Intense  electric  discharges 
through  a  non-metallic  gas  at  ordinary  pressures  or  in  vacuum 
tubes  give  line  or  band  spectra. 


£    C 


Fia.  581. 

732.  Band  Spectra  are  usually  composed  of  fine  lines,  as  shown 
in  Fig.  582  (the  spectrum  of  part  of  the  carbon  arc,  supposed  to  be 
due  to  cyanogen) .  The  light  of  the  green  cone  in  a  Bunsen  flame 
gives  a  very  similar  spectrum,  due  to  carbon  or  its  compounds. 
The  salts  of  the  calcium  group  of  metals  have  flame  spectra 
containing  both  lines  and  bands.  All  salts  of  calcium,  for  exam- 
ple, give  the  same  flame  spectrum  under  ordinary  conditions, 
but  if  calcium  chloride,  for  instance,  is  placed  in  a  fiame  supplied 
with  hydrochloric  acid  an  entirely  different  band  spectrum  is 


Fia.  582. 

produced,  and  still  another  if  the  flame  is  supplied  with  calcium 
bromide  and  an  excess  of  hydrobromic  acid.  The  inference  is 
that  in  these  cases  the  bands  represent  the  characteristic  spectra 
of  the  compounds,  and  that  the  spectrum  observed  under  ordi- 
nary conditions  is  that  of  the  oxide,  due  to  reaction  with  atmos- 
pheric oxygen. 

Nitrogen  gives  a  band  spectrum  very  similar  to  that  of  cyano- 
gen if  a  feeble  discharge  passes  through  it,  but  an  entirely  differ- 
ent line  spectrum  if  the  discharge  is  very  intense.  (Fig.  583.) 


EMISSION  OF  RADIANT  ENERGY 


639 


Nitric  oxide  gives  a  characteristic  band  spectrum  in  the  ultra- 
violet similar  to  that  of  nitrogen.  All  the  compounds  of  mercury 
with  chlorine,  bromine,  or  iodine,  give  characteristic  band  spectra 
with  feeble  discharges,  and  the  line  spectrum  of  mercury  with 
strong  discharges.  The  same  is  true  in  a  number  of  other  cases. 
All  these  facts  are  consistent  with  the  view  that  band  spectra 
are  characteristic  of  the  molecular  state  of  either  elements  or 
compounds.  Intense  discharges,  by  dissociating  the  molecule, 
will  produce  line  spectra,  characteristic  of  the  atomic  or,  rather, 
The  salts  of  the  alkali  metals  are  so  easily  dis- 


Fia.  583. 

sociated  that  they  give  only  line  spectra  in  the  flame.  It  is  pos- 
sible, however,  by  other  modes  of  excitation  to  produce  band 
spectra  of  these  elements  (see  §751). 

733.  Limits  of  the  Spectrum. — The  very  short  ultra-violet 
waves  are  absorbed  by  all  gases  except  hydrogen,  and  by  most 
lenses  and  prisms.  Working  with  fluorite  lenses  and  prisms  or 
a  grating  in  a  vacuum,  Schumann  and  Lyman  have  reached  a 
wave-length  of  about  .00006  mm.  The  ordinarily  used  unit  of 
wave-length  is  the  Angstrom  unit,  equal  to  one  ten-millionth  of 
a  millimeter.  This  is  sometimes  called  a  tenth-meter.  Another 
unit  frequently  used  is  the  micron,  fj.  —  0.001  mm.  Expressed  in 
these  units,  some  wave-lengths  are  given  below.  Most  substances 
are  opaque  to  very  long  waves,  and  some  of  the  longest  waves 
mentioned  were  obtained  by  the  method  of  selective  reflection 
described  in  §778,  the  wave-length  being  then  measured  by  a 
coarse  grating. 

It  has  been  proved  that  X-Rays  (§553)  and  gamma  rays 
(§575)  are  similar  to  light  waves,  but  very  much  shorter. 
Their  lengths  differ,  but  the  order  of  their  magnitude  is  given 
in  the  table  below. 


640  LIGHT 

Angstrdm  Units        fi 

Gamma  rays 0.1  0.00001 

X-rays 1  0.0001 

Shortest  ultra-violet  waves 600  .06 

Shortest  visible  waves  (violet),  about 3,800  0.38 

Violet,  about 4,000  0.4 

Blue 4,500  0.45 

Green 5,200  0.52 

Yellow 5,700  0.57 

Red 6,500  0.65 

Longest  visible  waves  (red) 7,500  0 . 75 

Longest  waves  in  solar  spectrum,  more  than 53,000  5 . 3 

Longest  waves  transmitted  by  fluorite 95,000  9.5 

Longest  waves  by  selective  reflection  from  rock  salt  500,000  50 . 0 

By  reflection  from  potassium  chloride 612,000  61 .2 

Longest  waves  from  mercury  lamp 3,140,000  314 

Shortest  electric  waves 40,000,000  =  4  mm.  4000 

734.  General  Absorption. — When  radiation  falls  on  matter  a 
portion  is  reflected,  another  absorbed,  and  if  the  substance  is 
transparent   or  very  thin  a   part  is  transmitted.     Black  sub- 
stances, such  as  lampblack  and  copper  oxide,  reflect  and  transmit 
very  little,  the  absorption  being  almost  complete.     Most  sub- 
stances black  to  visible  radiation  are  also  black  to  the  ultra- 
violet and  infra-red  waves,  but  there  may  be  exceptions — for 
example,  a  sheet  of  hard  black  rubber  is  opaque  to  visible  radia- 
tion, but  transparent  to  waves  beyond  the  red.     Substances 
like  that  last  mentioned,  which  absorb  certain  radiations  and 
transmit  others,  are  said  to  exercise  selective  absorption. 

735.  Selective  Absorption  is  characteristic  of  most  substances. 
Familiar  examples  are  red  glass,  which  transmits  red  and  some 
infra-red,  but  no  other  visible  colors;  blue  cobalt  glass,  which 
transmits  blue  and  violet  and  a  little  red  and  green  in  narrow 
regions;  green,  which  transmits  almost  all  the  colors,  but  a 
larger  proportion  of  green;  chlorophyll  solution,  potassium  per- 
manganate, the  aniline  colors,  and  solutions  of  the  rare  earths, 
didymium,  etc.     In  most  cases  the  absorption  bands  are  wide 
and  diffuse;  in  the  case  of  the  rare  earths  they  are  almost  as 
narrow  as  spectral  lines,  so  that  the  solutions  appear  almost 
colorless,  no  large  amount  of  any  one  color  being  absorbed;  the 
vapors  of  iodine,  nitrogen  peroxide,  and  some  other  substances 
have  fluted  absorption  bands,  grouped  somewhat  like  the  lines 


EMISSION  OF  RADIANT  ENERGY  641 

in  the  nitrogen  bands.  Many  substances  such  as  glass,  quartz 
and  rock  salt  are  very  transparent  within  wide  limits,  beyond 
which  they  are  completely  opaque. 

o 

Glass  is  opaque  to  waves  shorter  than  3500  AngstrSm  units,  and  longer 
than  about  30,000  AngstrQm  units.  Quartz  is  transparent  between  the 
wave-lengths  1800  and  70,000,  and  for  some  longer  waves;  rock  salt  s 
transparent  between  1800  and  180,000,  and  fluorite,  one  of  the  most  trans- 
parent substances,  will  transmit  ultra-violet  waves  from  about  X  =  1000  to 
X  =  95,000. 

736.  Kirchhoff 's  Law. — If  the  fraction  A  of  the  radiation  of  a 
given  wave-length  incident  on  a  body  is  absorbed,  A  is  said  to  be 
its  absorbing  power  for  that  color.     The  emissivity  of  a  radiat- 
ing body  is  the  amount  of  energy  radiated  per  second  from  each 
unit  of  surface.     Kirchhoff  showed  by  the  theory  of  exchanges 
(§335)  that  the  emissive  and  absorptive  powers  of  all  bodies  at 
the  same  temperature  for  a  given  color  are  proportional  when 
the  radiation  is  a  pure  temperature  effect. 

737.  Origin  of  the  Fraunhofer  Lines.    A  general  account  of 
these  lines  has  been  given  in  §673. 

Kirchhoff,  noting  that  there  were  coincidences  between  many 
of  the  Fraunhofer  lines  and  emission  lines,  explained  them  as 
the  result  of  absorption  by  vapors  in  the  sun's  atmosphere  of 
waves  which  these  vapors  emit  themselves.  Stokes  independently 
suggested  that  the  coincidence  of  the  yellow  sodium  lines  with  the 
D  lines  indicated  that  the  sodium  atoms  must  absorb  waves  of 
the  same  frequency  as  those  emitted  by  them,  the  effect  being 
similar  to  resonance  phenomena  in  sound.  This  reversal  of  the 
sodium  lines  is  easily  secured  by  igniting  a  small  piece  of  metallic 
sodium  in  a  metal  spoon  before  a  slit  illuminated  with  the  electric 
arc,  the  light  then  passing  through  a  prism  and  a  lens  which 
focuses  it  on  a  screen.  If  a  large  quantity  of  sodium  vapor  is 
present  in  an  arc  the  phenomenon  of  self-reversal  is  shown  in  the 
spectrum.  The  bright  lines  are  very  broad  and  intense,  with  a 
narrow  dark  line  in  the  middle  of  each,  due  to  absorption  by  the 
cooler  sodium  vapor  in  the  outer  portion  of  the  arc. 

738.  Luminescence. — In  all  cases  where  radiation  is  purely  a 
temperature  effect  Kirchhoff's  law  appears  to  hold.     In  many 
cases,  such  as  those  of  fluorescence  and  phosphorescence  (§§750, 
751),  in  which  the  absorption  of  waves  of  certain  lengths  causes 

41 


642 


LIGHT 


the  emission  of  waves  of  a  different  length,  this  is  not  true;  nor 
is  it  generally  true  of  luminous  gases  and  vapors,  where  the  lumi- 
nosity appears  to  be  due  to  electrical  or  chemical  causes.  In  no 
known  case  do  gases  or  vapors  have  absorption  lines  correspond- 
ing to  all  the  emission  lines.  The  name  luminescence  has  been 
applied  to  the  various  kinds  of  radiation  not  directly  due  to  high 
temperature  and  not  conforming  to  Kirchhoff's  law. 

739.  Solar  Spectrum. — The  wave-lengths  of  many  thousands  of 
the  Fraunhofer  lines  were  determined  by  Rowland.  A  large 
number  were  found  to  coincide  with  the  emission  lines  of  known 
elements,  so  that  it  seems  certain  that  about  forty  of  these  ele- 
ments exist  in  the  sun.  (See  Fig.  581,  which  shows  the  coinci- 
dence of  many  of  these  absorption  lines  with  the  emission  lines 
of  iron).  The  chromosphere,  or  gaseous  solar  atmosphere,  the 
prominences  or  flames  of  incandescent  hydrogen  and  other  gases 
rising  out  of  it,  and  the  corona,  or  nebulous  outer  envelope,  give 
bright  line  spectra,  which  may  be  seen  during  a  total  eclipse, 
when  the  brighter  light  from  the  photosphere  does  not  mask  them. 
The  rare  gas  helium  was  known  to  exist  in  the  sun  before  it  was 
found  on  the  earth,  on  account  of  the  bright  yellow  line  due  to  it 
observed  in  the  spectrum  of  the  prominences. 

The  ultra-violet  region  of  the  solar  spectrum  does  not  extend  beyond  a 
wave-length  of  about  3000  AngstrOm  units.  Without  doubt  shorter  waves 
are  emitted,  but  they  are  absorbed  by  the  earth's  atmosphere,  which  is 
opaque  to  all  very  short  waves.  The  atmosphere  also  exercises  general 
and  selective  absorption  in  the  visible  region.  Oxygen  and  water  vapor 
give  rise  to  the  terrestrial  lines  and  bands  known  as  the  Fraunhofer  lines 
A,  a  and  B,  and  there  is  more  or  less  general  absorption  due  to  these  and 
other  constituents  of  the  earth's  atmosphere. 

Wave-lengths  of  Fraunhofer  Lines 


A 

7594-7621 

0 

Red 

B 

6870 

0 

Red 

D, 

5896.15 

Na 

Orange 

D2 

5890.18 

Na 

Orange 

E3 

5269.72 

Fe 

Green 

F 

4861.50 

H 

Blue 

g 

4226.89 

Ca 

Violet 

H 

3968.62 

Ca 

Violet 

K 

3933.81 

Ca 

Violet 

EMISSION  OF  RADIANT  ENERGY  643 

The  infra-red  region  of  the  solar  spectrum  has  been  investigated  by 
Langley  with  the  bolometer,  and  found  to  extend  beyond  a  wave-length  of 
53,000  Angstrom  units.  Broad  absorption  bands  are  found,  some  of  which 
coincide  with  those  due  to  water  vapor  and  carbon  dioxide,  besides  many 
narrow  lines  and  bands  of  unknown  origin.  A  large  proportion  of  the  solar 
radiation,  particularly  in  the  neighborhood  of  the  shorter  waves,  is  absorbed 
by  the  earth's  atmosphere,  and  this  must  greatly  influence  climatic 
conditions. 

740.  Spectra   of   Planets,   Stars,   Comets,    and   Nebulae. — The 

planets  and  the  moon  give  spectra  similar  to  that  of  the  sun,  as 
might  be  expected,  but  modified  by  general  and  selective  ab- 
sorption in  the  cases  of  the  planets  which  have  an  atmosphere. 
Most  stars  have  characteristic  absorption  spectra  resembling 
that  of  the  sun,  which  shows  the  universal  distribution  of  many 
of  the  common  elements.  In  addition  there  are  frequently 
lines  due  to  unknown  elements.  Nebulse  give  bright  line  spectra, 
some  of  the  lines  being  due  to  hydrogen  and  helium,  while  others 
have  not  yet  been  identified.  The  spectrum  of  comets  consists 
mostly 'of  the  characteristic  hydrocarbon  bands  similar  to  those 
given  by  the  green  cone  of  the  Bunsen  flame.  It  seems  evident 
in  the  cases  of  nebulae  and  comets  that  the  radiation  is  an  ex- 
ample of  luminescence,  or  luminosity  due  to  other  causes  than 
high  temperature,  because  these  bodies  appear  to  consist  of 
masses  of  highly  attenuated  gases,  or  small  bodies,  and  it  is 
inconceivable  that  their  temperature  can  permanently  remain 
much  higher  than  that  of  the  surrounding  space. 

741.  Application  of  Doppler's  Principle. — If  a  star  is  approach- 
ing or  receding  from  the  earth,  the  effect  will  be  to  shorten  or 
lengthen  each  wave  reaching  the  earth  ( §596) .     Each  line  will  be 
displaced  toward  the  violet  if  the  star  is  approaching,  toward  the 
red  if  it  is  receding.     By  measuring  such  displacements  on 
photographs  of  stellar  spectra  the  velocities  of  stars  in  the  line  of 
sight  may  be  determined  with  an  error  of  less  than  one  kilometer 
per  second.     Most  of  the  stars  which  have  been  investigated 
have  velocities  with  respect  to  the  sun  of  between  one  and  one 
hundred  kilometers  per  second.     It  is  found  that  a  majority  of 
the  stars  on  one  side  of  the  heavens  have  a  general  relative 
motion  toward  the  sun,  those  on  the  opposite  side  away  from 
the  sun.     The  inference  is  that  the  solar  system  is  itself  moving 
through  the  universe  in  the  former  direction . 


644  LIGHT 

Fig.  584  is  the  spectrum  of  ^  Draconii,  with  comparison  spec- 
tra of  hydrogen  above  and  below,  showing  the  Doppler  effect  on 
the  hydrogen  absorption  lines. 


|Fia.  584. 

EFFECTS  DUE  TO  ABSORPTION 

742.  Color  of  Natural  Objects. — The  colors  seen  in  the  spectra 
produced  by  dispersion  or  by  interference  are  pure.     This  is  not 
the  case  with  the  colors  of  natural  objects,  which  as  a  rule  are 
due  to  selective  absorption  of  certain  colors  of  the  incident  light, 
the  other  colors  being  diffusely  reflected  in  different  proportions. 
If  a  colored  object,  such  as  a  red  rose,  is  placed  in  different 
parts  of  a  spectrum,  it  will  appear  a  brilliant  red  in  the  red  and 
almost  black  in  other  parts.     This  shows  that  the  greater  part  of 
all  colors  except  red  is  absorbed;  not  all,  however,  for  it  will  be 
noticed  that  in  every  part  of  the  spectrum  there  is  some  reflec- 
tion of  the  incident  color.     Since  the  resultant  of  the  combina- 
tion of  all  colors  is  white,  it  may  thus  be  proved  that  from  all 
colored  objects  some  white  light  is  reflected,  in  addition  to  the 
characteristic  color. 

743.  Body  Color. — In  most  cases  it  is  observed  that  bodies 
having  a  certain  color  by  reflected  light  have  the  same  color  by 
transmitted  light.     This  suggests  that  the  color  diffusely  reflected 
is  due  to  components  of  the  incident  white  light  which  have  pene- 
trated more  or  less  into  the  medium  before  being  scattered,  the 
other  colors  being  lost  by  absorption.     The  white  light  reflected 
is  probably  due  both  to  reflection  at  the  surface  and  to  the  re- 
combination of  the  various  colors  which  escape  complete  absorp- 
tion.    As  a  crude  illustration  of  body  color,  if  light  falls  on  a 
piece  of  red  glass  a  white  image  of  the  source  will  be  reflected 
from  the  front  surface  and  a  red  image  from  the  rear  surface. 

Colors  are  said  to  be  more  or  less  saturated  according  to  the 
proportion  of  white  light  with  which  they  are  diluted.  The  pure 
spectral  colors  are  said  to  be  completely  saturated.  The  pro- 


EFFECTS  DUE  TO  ABSORPTION  645 

portion  of  white  light  scattered  is  increased  by  any  process  which 
increases  the  reflecting  surface.  For  example,  crystals  of  copper 
sulphate  will  appear  lighter  and  lighter  as  they  are  crushed  into 
smaller  fragments,  and  become  almost  white  when  reduced  to  a 
fine  powder.  The  white  light  reflected  from  the  numerous  sur- 
faces then  completely  masks  the  small  portion  which  is  select- 
ively transmitted.  Similarly,  transparent  substances  such  as 
glass  are  white  when  in  powdered  form. 

744.  Dichromatism. — Some   substances   when   examined    by 
light  transmitted  through  thick  layers  appear  to  be  of  different 
color  from  that  observed  by  reflection  or  by  transmission  through 
a  thin  layer.     A  thin  layer  of  chlorophyll  is  green  by  transmitted 
light,  while  a  thick  layer  is  red.     This  is  explained  by  the  fact 
that  the  absorptive  power  or  the  fraction  of  the  incident  light 
absorbed  by  a  layer  of  unit  thickness,  is  different  for  the  two 
colors.     While  the  incident  green  light  is  more  intense  than 
the  red,  and  remains  so  after  transmission  through  a  thin  layer, 
it  is  more  rapidly  cut  down  by  absorption,  so  that  after  passing 
through  a  thick  layer  the  red  predominates.     This  effect  is 
called  dichromatism. 

745.  Surface   Color. — Some   substances   appear   of   different 
colors  by  reflected  and  by  transmitted  light.     Such  is  the  case 
with  thin  films  of  metal  and  of  the  solid  aniline  colors.     Gold  is 
always  yellow  by  reflected  light,  but  a  sheet  of  gold  leaf  thin 
enough  to  permit  transmission  appears  green  by  the  transmitted 
light.    The  light  reflected  from  these  substances  is  complementary 
to  that  transmitted.     In  such  cases  selective  action  seems  to 
take  place  at  the  surface,  some  colors  being  directly  reflected, 
others  being  absorbed  by  a  thick  layer,  or  transmitted  through  a 
thin  film.     Bodies  exhibiting  surface  color  retain  that  color  when 
finely  powdered. 

746.  Colors  of  Sky  and  Clouds. — Since  light  can  reach  the  eye 
only  directly  from  the  source  or  by  reflection  from  material 
objects,  it  is  evident  that,  since  the  sky  is  not  perfectly  black,  it 
must  contain  matter  in  suspension.     Some  have  supposed  that 
air  itself  may  have  a  characteristic  color,  as  is  shown  by  great 
thicknesses  of  glass  or  of  water,  but  it  is  probable  that  the 
blue  color  of  the  sky  is  due  to  selective  scattering  by  small 
suspended  particles  of  dust,  water,  etc.     It  is  to  be  expected 


646  LIGHT 

that  such  small  particles  should  reflect  a  larger  proportion  of 
short  waves  than  of  long  ones.  The  term  scattering  is  used, 
because  it  seems  evident  that  this  is  not  a  case  of  ordinary 
reflection  like  that  from  a  mirror  of  finite  size.  There  is  an 
analogy  in  the  case  of  sound  waves;  long  waves  pass  around 
obstacles  without  deviation  from  their  general  direction,  while 
shorter  waves  may  be  reflected.  Since  the  shorter  waves  of  light 
are  scattered,  the  transmitted  light  will  consist  mostly  of  the 
longer  waves.  This  accounts  for  the  brilliant  reds,  oranges,  and 
greens  often  observed  in  the  western  sky  at  sunset.  The  light 
transmitted  almost  tangentially  through  the  atmosphere  has  been 
deprived  of  the  shorter  waves,  which  cause  a  blue  sky  for  those 
more  immediately  under  the  sun.  These  effects  are  intensified 
by  the  presence  of  a  large  number  of  dust  particles  in  the  lower 
levels  of  the  atmosphere.  After  the  great  eruption  of  the  volcano 
Krakatoa  in  1883  fine  volcanic  dust  pervaded  the  atmosphere  of 
the  whole  earth  and  the  sunsets  were  especially  brilliant.  For 
the  same  reason  lights  look  red  when  seen  through  smoke  or  fog, 
or  through  water  made  slightly  turbid  by  the  addition  of  a  small 
quantity  of  milk  or  shellac  solution.  This  effect  is  beautifully 
illustrated  by  passing  a  beam  of  light  through  a  jet  of  steam 
issuing  from  a  small  nozzle  into  a  stream  of  air  previously  dried 
by  forcing  it  through  sulphuric  acid.  The  size  of  the  water  drops 
is  controlled  by  changing  the  vapor  pressure  in  the  atmosphere  in 
which  the  drops  are  formed,  lower  vapor  pressure  promoting 
evaporation  and  thus  reducing  the  size  of  the  drops.  The  colors 
seen  by  transmitted  and  by  scattered  light  are  complementary, 
the  shorter  waves  being  scattered  and  the  longer  ones  trans- 
mitted. 

747.  Color  Sensation. — The  perception  of  a  given  color  by  the 
eye  does  not  necessarily  prove  that  the  stimulus  is  of  the  corre- 
sponding wave-length.  It  may  be  the  resultant  effect  of  several 
different  colors.  For  example,  if  the  light  from  the  red  of  a 
spectrum  and  from  a  region  intermediate  between  the  blue  and 
the  green  be  superimposed  the  resultant  sensation  is  white, 
which  the  eye  cannot  distinguish  from  the  white  due  to  a  mixture 
of  all  the  colors.  A  similar  effect  is  produced  by  the  combination 
of  violet  and  yellow-green.  Two  colors  which  together  give  the 
sensation  of  white  are  said  to  be  complementary.  It  is  found, 


EFFECTS  DUE  TO  ABSORPTION  647 

further,  that  spectral  red  and  green  combined  excite  a  sensation 
of  yellow,  while  green  and  violet  produce  blue.  All  possible 
colors  may  be  produced  by  combining  red,  green,  and  violet. 
According  to  the  theory  of  Thomas  Young,  these  are  to  be 
regarded  as  the  three  primary  color  sensations.  The  cones  in  the 
retina  are  supposed  to  respond  or  "resonate"  most  actively  to 
frequencies  of  vibration  corresponding  to  these  colors,  and  all 
color  sensations  depend  on  the  proportions  of  the  incident  energy 
belonging  to  these  frequencies. 


3.  Violet 


K   H      6  F  ED  C        B 

Fio.  585. 


The  phenomena  of  color  sensation  may  be  explained  by  assuming  that 
in  the  normal  eye  there  are  three  sets  of  nerves,  one  stimulated  most  actively 
by  red  light,  another  by  green,  and  a  third  by  violet,  but  each  responding 
also  more  or  less  to  waves  of  other  frequencies  as  well.  To  indicate  these 
effects  Koenig  constructed  three  curves  (Fig.  585)  on  an  axis  representing 
the  length  of  the  normal  spectrum  from  the  Fraunhofer  line  K  in  the  violet 
to  B  in  the  red.  The  ordinates  at  each  point  represent  the  degree  of  excita- 
tion of  the  three  sets  of  nerves  respectively  by  light  of  the  frequency  corre- 
sponding to  that  point  in  the  spectrum.  The  maximum  sensibility  of  the 
"red"  nerves  is  in  the  orange-red,  that  of  the  "green"  nerves  in  the  green, 
and  that  of  the  "violet"  nerves  in  the  blue- violet.  The  first  set  of  nerves 
is  also  excited  more  or  less  by  all  colors  between  H  and  B ;  the  green  by  all 
colors  between  G  and  C,  and  the  violet  by  all  colors  between  K  and  E.  The 
color  of  sodium  light  (D)  is  caused  by  the  superposition  of  two  sensations, 
red  and  green,  proportional  respectively  to  the  ordinates  Dl  and  D2.  The 
color  of  the  blue  line  of  hydrogen  (F)  is  due  to  a  combination  of  red,  green, 
and  violet  sensations  proportional  to  the  ordinates  Fl,  F2,  and  F3.  In  the 
case  of  color-blind  persons  one  or  more  sets  of  nerves  are  missing — usually 
the  red.  To  such  persons,  for  example,  sodium  light  would  appear  green. 
Two  colors,  such  as  red  and  blue-green,  are  complementary,  when,  acting 
jointly,  they  excite  all  three  sets  of  nerves  in  the  proper  proportion  to  pro- 
duce the  sensation  of  white — or  in  the  same  proportions  that  they  are 
excited  by  ordinary  white  light. 


648  LIGHT 

748.  Pigment  Colors. — The  effect  of  mixing  pigments  is  quite 
different  from  that  of  mixing  spectral  colors.     For  example,  blue 
paint  absorbs  nearly  all  the  incident  light  except  the  blue  and  some 
green;  yellow  paint  absorbs  nearly  all  except  yellow  and  some 
green.     If,  therefore,   white  light  is  incident  on  a  mixture  of 
the  two  pigments  green  is  the  only  color  which  escapes  absorption 
by  one  or  the  other,  therefore  a  mixture  of  blue  and  yellow  paints 
produces  a  green  paint.     In  such  cases  the  apparent  color  may 
vary  with  the  kind  of  illumination.     Blue  pigments  usually 
appear  green  by  candle  light,  because  there  is  a  very  small  propor- 
tion of  blue  in  the  incident  light,  and  so  green  predominates  in  the 
scattered  light. 

749.  Chemical  and  Molecular  Effects. — Light  may  cause  chemical  com- 
bination, as  when  it  acts  on  a  mixture  of  hydrogen  and  chlorine,  or  dis- 
sociation, as  when  it  acts  on  the  silver  salts  in  a  photographic  plate.     By 
its  action  on  the  chlorophyll  of  plants,  light  decomposes  the  carbon  dioxide 
absorbed  from  the  atmosphere,  releasing  the  oxygen  and  causing  the  carbon 
to  be  assimilated.     It  may  cause  molecular  transformations,  as  when  it 
alters  amorphous  to  crystalline  selenium,  or  changes  the  electric  resistance  of 
the  latter  form.     It  also  changes  white  phosphorus  to  red.     These  effects 
are  not  due  to  the  heating  effect  of  the  absorbed  radiation,  because  an  equiva- 
lent rise  of  temperature  will  not  cause  them,  but  they  seem  rather  to  depend 
on  the  vibratory  character  of  the  light  waves.     As  a  rule  the  shorter  waves 
are  the  most  effective  in  producing  such  results. 

Another  effect  due  to  light,  especially  to  the  ultra-violet  waves,  is  that  it 
will  cause  the  discharge  of  electricity  from  certain  metals  (§565). 

750.  Fluorescence. — There  are  substances  which  when  stimu- 
lated by  the  absorption  of  waves  of  certain  lengths  will  emit  waves 
of  different  lengths.  For  example,  a  piece  of  paper  moistened 
with  sulphate  of  quinine  solution  and  held  in  the  ultra-violet 
portion  of  the  solar  spectrum  will  emit  a  brilliant  opalescent  blue 
light.  To  this  phenomenon  Stokes  gave  the  name  of  fluorescence, 
because  it  was  observed  in  fluorspar.  He  explained  it  as  the 
result  of  the  absorption  of  incident  waves  which  by  a  modified 
resonance  action  caused  a  regmission  of  longer  waves.  Similar 
effects  are  observed  in  coal  oil,  fluorescein,  eosin,  uranin,  and 
other  organic  compounds;  in  uranium  glass,  which  emits  a  yellow- 
ish-green light;  in  esculin,  which  emits  blue  light,  and  in  chloro- 
phyll, which  emits  red  light;  and  also  in  a  much  smaller  degree  in 
iodine,  wood,  paper,  and  many  other  substances. 


DOUBLE  REFRACTION  AND  POLARIZATION  649 

751.  Phosphorescence. — There  is  a  large  class  of  substances,  of 
which  calcium,  strontium,  and  barium  sulphides  are  familiar 
examples,  which  after  exposure  to  light  show  effects  which  are 
similar  to  fluorescence,  but  which  continue  visible  long  after  the 
exciting  radiation  ceases  to  act.     This  is  called  phosphorescence. 
The  only  definite  distinction  between  fluorescence  and  phosphor- 
escence is  that  the  latter  persists  for  a  longer  time.     Many  sub- 
stances which  phosphoresce  very  feebly  at  ordinary  temperatures 
may  be  made  to  glow  brilliantly  at  the  temperature  of  liquid  air. 
As  examples,  gelatin,  horn,  egg  shells,  and  paper  may  be  men- 
tioned. 

Some  metallic  vapors,  such  as  those  of  the  sodium  and  calcium 
group,  fluoresce  brilliantly  under  the  action  of  light  or  cathode 
rays..  The  light  shows  the  characteristic  spectral  lines  and  bands 
of  the  metal.  Certain  organic  vapors,  such  as  anthracene,  fluo- 
resce when  light  falls  on  them.  Nitrogen,  oxygen,  and  some 
other  gases  will  under  certain  conditions  phosphoresce  brightly 
for  several  seconds  after  an  electric  discharge  has  passed  through 
them  in  a  vacuum  tube  (§550). 

DOUBLE  REFRACTION  AND  POLARIZATION 

752.  Double  Refraction. — Some  crystals,  such  as  those  of  rock 
salt  and  fluorite,  resemble  isotropic  solids,  such  as  glass,  in  the 
respect  that  their  physical  properties  are  alike  in  all  directions. 
In  general,  however,  this  is  not  the  case;  such  properties  as  elas- 
ticity and  heat  conduction,  as  well  as  optical  properties,  differ  in 
different  directions  in  the  crystal.     In  such  crystals  as  quartz  and 
calcite  there  is  an  axis  of  symmetry,  the  crystallographic  axis,  and 
the  physical  properties  are  the  same  in  all  directions  in  any 
equatorial  plane,  but  different  from  those  in  the  direction  of  the 
axis.     Iceland  spar,  or  calcite,  is  a  rhombohedral  crystal,  each 
face  being  a  parallelogram  with  two  acute  angles  of  78°  5'  and 
two  obtuse  angles  of  101°  55'.     Two  solid  angles  of  the  crystal  are 
formed  by  the  junction  of  the  obtuse  angles  of  three  faces.   Any 
line  equally  inclined  to  the  faces  of  one  of  these  solid  angles  is  a 
crystallographic   axis.     An  object  seen  through  Iceland  spar 
appears  double,  unless  viewed  in  the  direction  of  the  axis.    No 
such  effect  is  observed  in  the  case  of  isometric  crystals.     This 


650 


LIGHT 


phenomenon  is  called  double  refraction.  When  the  waves  travel 
in  the  crystal  in  the  direction  of  the  crystallographic  axis  there  is 
no  double  refraction;  hence  any  line  in  the  crystal  parallel  to  the 
axis  is  called  an  optic  axis. 

If  a  ray  r  of  ordinary  light  is  incident  normally  on  any  face  of  a 
doubly-refracting  crystal  one  ray  o  is  transmitted  without  devia- 
tion; and  if  the  incidence  is  oblique  (Fig.  586)  this  ray  is  deviated, 


00' 


with  an  index  of  refraction  which  is  independent  of  the  angle  of 
incidence.  The  other  ray  e  is  deviated  in  all  cases,  unless  it 
travels  along  an  optic  axis,  and  the  index  of  refraction  varies 
with  the  angle  of  incidence.  The  first  is  called  the  ordinary,  the 
second  the  extraordinary  ray.  If  the  crystal  be  rotated,  keeping 
the  angle  of  incidence  constant,  the  ordinary  image  will  remain  at 
rest,  while  the  extraordinary  image  rotates  about  it  in  such  a 
way  that  the  line  joining  the  two  images  lies  in  a  principal 
section,  a  plane  including  the  normal  to  the  surface  and  an  optic 
axis.  If  the  ordinary  and  extraordinary  rays  o  and  e  pass  through 
a  second  crystal  each  ray  generally  divides  in  two,  the  rays  oo' 
and  oe'  and  the  rays  eo'  and  ee'  (Fig.  586),  the  line  joining  each 
pair  lying  in  a  principal  section  of  the  second  crystal.  This 
gives  rise  to  four  images  of  the  source,  which  are  of  equal  in- 
tensity when  the  principal  sections  of  the  two  crystals  are  at  an 
angle  of  45°  with  each  other.  If  this  angle  be  changed  one  pair 
of  images  will  increase  in  intensity  and  the  other  diminish.  When 
the  principal  sections  are  parallel  only  the  rays  oo'  and  ee'  emerge; 
when  they  are  at  right  angles,  only  the  rays  oe'  and  eo'.  From 
such  experiments  Huyghens  recognized  the  fact  that  light  which 
has  passed  through  Iceland  spar,  quartz,  and  other  doubly 
refracting  crystals  does  not  possess  properties  which  are  alike  in 
all  azimuths  around  the  direction  of  propagation.  Newton,  in 


DOUBLE  REFRACTION  AND  POLARIZATION  651 

order  to  explain  this,  supposed  the  light  corpuscles  to  be  endowed 
with  polarity  of  some  sort — hence  the  name  polarized  light. 

753.  Direction  of  Vibration. — Fresnel  explained  the  phenome- 
non of  double  refraction  as  a  result  of  the  transverse  vibration  of 
light  waves.    If  the  vibrations  were  longitudinal,  it  is  impossible 
to  conceive  how  they  could  be  affected  by  rotation  of  the  crystal 
in  a  plane  at  right  angles  to  the  direction  of  propagation.     Trans- 
verse vibrations  in  a  cord  may  be  said  to  be  polarized.     Such 
vibrations  would  be  freely  transmitted  through  a  slot  parallel  to 
the  direction  of  vibration,  but  not  through  one  at  right  angles  to 
this  direction.       Longitudinal   vibrations   in  a  cord  could  be 
freely  transmitted  through  a  slot,   regardless   of   its   position. 
Fresnel  assumed  that  in  ordinary  white  light  successive  waves 
reaching  a  given  point  of  space  vibrate  in  different  planes  at 
random,  so  that,  although  each  individual  wave  is  vibrating 
transversely  in  a  definite  plane,  and  is,  therefore,  polarized,  this 
direction  changes  so  rapidly  that  the  eye  cannot  take  account 
of  it  and  no  polarization  effects  are  observed.     In  passing  through 
a  doubly-refracting  crystal  vibrations  in  one  direction  travel 
with  a  different  velocity  from  those  in  another  direction,  on 
account  of  the  difference  of  the  physical  properties  of  the  crystal 
in  these  directions,  hence  double  refraction  results.    The  displace- 
ment in  each  wave  is  in  general  resolved  into  two  components, 
unless  the  light  is  travelling  parallel  to  the  axis.    In  that  case  it  is 
unmodified,  as  the  velocity  of  propagation  is  independent  of  the 
azimuth.     In  the  ordinary  ray,  which  travels  in  all  directions 
with  the  same  velocity,  the  vibrations  must  be  at  right  angles  to 
the  optic  axis.     So  long  as  this  is  the  case  the  displacements  will 
take  place  under  the  same  conditions  in  every  azimuth  and  the 
velocity  be  unchanged.     In  the  extraordinary  ray  the  vibrations 
must  be  in  a  principal  section.     This  accounts  for  the  fact  that 
the  ordinary  and  the  extraordinary  image  are  always  in  a  line 
parallel  to  a  principal  section. 

754.  The  Wave  Surfaces. — From  the  experiments  described 
above  it  may  be  seen  that  a  wave  of  ordinary  light  on  entering  a 
doubly-refracting  crystal  is  divided  into  two  waves,  one  of  which, 
o,  has  the  same  velocity  in  all  directions  in  the  crystal.    The  other 
wave  ehas  a  velocity  which  varies  in  different  directions,  and  is  the 
same  as  that  of  the  ordinary  wave  only  when  both  travel  in  the 


652 


LIGHT 


direction  of  the  optic  axis.  Huyghens  showed  that  these  facts  are 
consistent  with  the  existence  of  a  double  wave  surface  in  the  crystal, 
a  sphere  and  an  ellipsoid  of  revolution,  which  are  tangent  to  each 
other  at  the  two  points  where  they  intersect  an  optic  axis.  In 
one  class  of  crystals,  like  Iceland  spar,  the  sphere  is  inside  the 
ellipsoid,  and  the  ordinary  wave  is  the  more  refracted  (Fig.  587). 
In  another  class,  represented  by  potassium  sulphate  or  quartz, 


Fio.  587. 


Fio.  588 


the  sphere  encloses  the  ellipsoid,  and  the  ordinary  ray  is  less 
refracted  (Fig.  588) .  The  first  are  called  negative  and  the  second 
positive  crystals.  In  the  case  of  quartz  the  two  wave  surfaces  do 
not  touch  where  they  intersect  the  optic  axis,  and  there  is  double 
refraction  of  another  kind  in  the  direction  of  the  axis  (§771). 
In  crystals  in  which  the  physical  properties  are  different  along 
three  axes  at  right  angles  to  each  other,  such  as  sugar  and  topaz, 
which  likewise  show  double  refraction,  there  are  two  axes  of  no 
double  refraction;  hence  such  crystals  are  said  to  be  biaxial,  as 
contrasted  with  the  class  described  above,  which  are  said  to  be 
uniaxial.  Both  rays  in  biaxial  crystals  are  extraordinary,  that 
is  to  say,  do  not  conform  to  the  ordinary  laws  of  refraction. 

755.  Double  Refraction  by  Tourmaline. — Tourmaline  is  a  semi- 
transparent  hexagonal  crystal.     If  light  falls  on  a  crystal,  part  is 

transmitted.  If  this  falls  on  a 
second  plate  with  its  axis  parallel 
to  that  of  the  first  (Fig.  589) ,  some 
of  the  light  gets  through;  but  if 
the  second  crystal  is  rotated 
about  the  line  joining  the  two, 
less  light  gets  through,  and  when 
its  axis  is  at  right  angles  to  that  of  the  first  none  is  transmitted. 
Evidently  the  waves  have  had  their  mode  of  vibration  so 
changed  by  passage  through  the  first  plate  that  they  cannot 
pass  through  the  second  unless  the  principal  sections  of  the  two 


589. 


DOUBLE  REFRACTION  AND  POLARIZATION 


653 


are  parallel.  If  the  light  first  passes  through  Iceland  spar  it 
is  found  that  the  extraordinary  ray  alone  will  pass  through 
tourmaline  if  the  principal  sections  of  the  two  crystals  are 
parallel,  the  ordinary  ray  alone  if  they  are  at  right  angles.  It 
follows  that  light  is  doubly  refracted  by  tourmaline,  but  that 
the  ordinary  ray  is  totally  absorbed. 

As  a  remarkable  example  of  Kirchhoff's  law  (§736),  it  may  be  mentioned 
that  if  tourmaline  is  raised  to  a  high  temperature  it  emits  polarized  radiation. 
If  this  falls  on  a  second  crystal  parallel  to  the  first  it  is  absorbed,  showing 
that  it  corresponds  to  the  ordinary  ray.  The  mode  of  vibration  which  is 
absorbed  corresponds  to  that  which  is  emitted. 

756.  Polarization  by  Reflection.— About  1808  Malus  discovered 
that  light  reflected  from  glass  at  a  definite  angle  acquires  prop- 
erties similar  to  that  of  light 
transmitted  through  tourma- 
line or  Iceland  spar.  When 
light  is  polarized  by  reflection 
from  a  mirror  A  (Fig.  590a)  a 
large  fraction  is  reflected  from 
another  mirror  B  if  the  two 
planes  of  incidence  coincide. 
If  the  planes  of  incidence  are 
at  right  angles  (Fig.  5906)  very 
little  is  reflected.  If  the  light 

reflected  from  a  glass  plate  is  examined  through  a  crystal  of 
Iceland  spar,  the  ordinary  ray  alone  is  transmitted  when  the 
plane  of  reflection  coincides  with  a  principal  section,  the  ex- 
traordinary ray  alone  when  the  two  are  at  right  angles.  In 
intermediate  positions  portions  of  both  rays  are  transmitted. 
Similarly  light  reflected  from  glass  is  not 
transmitted  through  tourmaline  if  the  plane 
of  reflection  is  parallel  to  the  optical  axis  of 
the  crystal. 

The  simplest  explanation  of  these  effects 
seems  to  be  that  when  light  strikes  a  re- 
flecting surface  there  is  a  partial  resolution  into  components 
respectively  in  and  at  right  angles  to  the  plane  of  incidence. 
The  vibrations  parallel  to  the  surface  are  most  freely  reflected, 


FIG    590. 


\ 


FIQ.  591. 


654  LIGHT 

while  the  others  strike  down  into  the  surface  and  are  transmitted 
or  absorbed  (Fig.  591).  If  polarized  light  is  incident  at  the 
angle  of  maximum  polarization  (§759)  on  a  piece  of  glass  a 
large  proportion  will  be  reflected  when  its  vibrations  are 
parallel  to  the  surface;  if  the  vibrations  are  in  the  plane  of  inci- 
dence it  will  be  refracted.  In  general  both  components  are 
reflected  and  refracted,  but  the  reflected  light  contains  a  larger 
proportion  of  waves  vibrating  perpendicularly  to  the  plane  of 
incidence,  and  the  refracted  light  a  larger  proportion  of  the 
waves  vibrating  parallel  to  that  plane. 

It  is  clear  that  no  interference  effects  can  be  produced  between 
two  vibrations  in  planes  at  right  angles  to  each  other. 

This  fact  enabled  Wiener  to  determine  the  direction  of  vibration  in  light 
polarized  by  reflection.  A  beam  polarized  by  the  mirror  M  fell  at  an  angle 

of  45°  on  a  thin  transparent  photo- 
graphic film  above  a  reflecting  sur- 
face. He  found  that  stationary 
waves  (§700)  were  produced  when 
the  plane  of  incidence  on  the  film 
coincided  with  the  plane  of  reflection 
from  the  mirror  (Fig.  592A),  but  this 
was  not  the  case  if  the  two  planes 
were  at  right  angles  to  each  other 
(Fig.  5925).  From  the  figure  it  ap- 
pears that  in  the  first  case  the  vibra- 
592-  tions  must  have  been  parallel  to  the 

film,  and  therefore  to  the  mirror,  in 

order  that  the  incident  and  reflected  rays  should  be  in  a  condition  to  inter- 
fere at  P,  while  in  the  second  case  the  vibrations  must  have  been  in  the 
plane  of  incidence  on  the  film,  and  therefore  parallel  to  the  mirror,  in 
order  that  the  vibrations  should  meet  at  P  at  right  angles  to  each  other. 
This  demonstrates  that  the  vibrations  in  light  polarized  by  reflection  are 
parallel  to  the  mirror. 

757.  Plane  Polarized  and  Ordinary  Light. — The  experimental 
evidence  warrants  the  assumption  that  light  waves  are  excited 
by  the  vibrations  of  the  particles  of  material  sources,  these 
particles  being  probably  ions  or  electrons  within  the  molecules 
(§729);  that  these  particles  in  general  vibrate  in  different 
planes  and  directions,  and  that  the  vibrations  of  a  given  particle 
may  constantly  change  in  direction;  that  each  vibrating  particle 
sends  out  into  the  surrounding  medium  a  series  of  waves  vibrating 
in  the  same  plane  as  the  particle,  so  that  ordinary  white  light 


DOUBLE  REFRACTION  AND  POLARIZATION  655 

consists  of  a  mixture  of  waves  of  many  lengths,  the  resultant 
vibrations  being  in  a  plane  at  right  angles  to  the  direction  of 
propagation,  successive  trains  of  waves  having  different  planes 
of  vibration;  and  that  by  double  refraction  or  reflection  we  may 
sift  out  component  vibrations  in  a  given  plane,  and  produce 
what  is  called  polarized  light.  When  all  the  vibrations  are  in 
parallel  planes  the  light  is  said  to  be  plane  polarized.  If  such  light 
is  mixed  with  ordinary  light,  it  is  said  to  be  partially  polarized. 
If  phase  differences  are  introduced  between  two  vibrations  at 
right  angles,  the  resultant  displacement  may  be  elliptical  or 
circular  (§243.)  This  gives  rise  to  elliptically  or  circularly 
polarized  light. 

758.  Plane  of  Polarization. — Before  the  direction  of  vibration 
in  polarized  light  was  known  it  became  customary  to  speak  of 
the  "plane  of  polarization"  of  a  polarized  beam,  rather  than  of 
the  direction  of  vibration,  and  this  plane  was  so  defined  that  it 
coincides  with  the  plane  of  incidence  when  the  light  is  polarized 
by  reflection.  It  follows  that  the  vibrations  in  a  polarized  beam 
are  at  right  angles  to  the  plane  of  polarization. 

769.  Brewster's  Law. — The  light  reflected  from  a  surface  is  not  in  general 
completely  polarized,  that  is,  all  its  vibrations  are  not  strictly  in  one  plane. 
It  is  found,  however,  that  for  each  reflecting  substance  there  is  a  certain 
angle  of  incidence  for  which  the  polarization  is  a  maximum.  This  is  called 
the  polarizing  angle.  It  was  found  by  Fresnel  that  complete  polarization 
is  given  only  by  substances  having  an  index  of  refraction  equal  to  about  1.46. 
Brewster  found  that  the  polarizing  angle  is  such  that  the  reflected  and  the 
refracted  rays  are  at  right  angles  to  each  other.  Since  n  —  (sin  i)  /  (sin  r)  and 
since,  when  i  =  p,  the  polarizing  angle,  p  +  r  =  90°, 

n  —  (sin  p)  I  (cos  p)  -» tan  p 

From  this  relation  the  polarizing  angle  p  may  be  determined.  This  is  known 
as  Brewster's  law. 

When  the  angle  of  incidence  is  different  from  that  defined  by  this  re- 
lation, and  even  for  that  angle  when  the  index  of  refraction  differs 
appreciably  from  1.46,  a  small  part  of  the  component  at  right  angle  to  the 
plane  of  incidence  is  reflected,  with  a  phase  different  from  that  of  the  other 
component,  resulting  in  elliptically  polarized  light  (§769). 

760.  Pile  of  Plates. — Since  only  a  small  fraction  of  the  incident 
light  is  reflected  from  a  transparent  substance,  even  when  the 
reflected  light  is  completely  polarized,  that  which  is  refracted  will 
be  only  partially  polarized;  that  is  to  say,  along  with  light  vibrat- 


656 


LIGHT 


ing  in  the  plane  of  incidence  a  considerable  proportion  of  that 
vibrating  at  right  angles  to  this  plane  will  be  transmitted.  If  it 
is  subject  to  a  second  reflection  the  proportion  of  polarized  light 
is  increased.  After  passing  through  eight  or  ten  plates  the 
transmitted  light  is  almost  completely  polarized.  If  a  pile  P 
of  thin  glass  plates  is  built  up  as  shown  in  Fig.  593,  the  beam  R, 
the  result  of  successive  reflections,  and  the  beam  T,  which  is 
transmitted,  are  completely  polarized  in  planes  at  right  angles 
to  each  other.  This  is  one  of  the  simplest  methods  of  securing 
polarized  light. 


FIQ.  593. 


FIG.  594. 


761-  Wave  Front  Construction. — If  C  is  a  radiant  point  in  a  crystal  of 
Iceland  spar  (Fig.  594)  and  if  A  A'  is  the  optic  axis  passing  through  that 
point,  two  waves  will  diverge  from  C,  one  spherical  and  the  other  spheroidal 
These  waves  will  have  the  same  velocity  along  A  A',  but  in  other  directions 
the  extraordinary  wave  will  travel  faster  than  the  ordinary.  The  vibrations 
in  each  wave  will  be  in  the  wave  surface.  The  vibrations  in  the  ordinary 
ray  will  everywhere  be  at  right  angles  both  to  the  optic  axis  and  to  the  direc- 
tion of  propagation.  In  the  extraordinary  wave  the  vibrations  are  in 
general  oblique  both  to  the  optic  axis  and  to  the  direction  of  propagation  of 
the  disturbance.  In  this  case  we  have  an  exception  to  the  general  rule  that 
the  wave  normal  indicates  the  direction  of  propagation. 

By  the  application  of  Huyghens'  principle  the  wave  fronts  in  double 
refraction  may  easily  be  determined.  Consider  a  plane  wave  A B  incident 
on  a  crystal  so  cut  that  the  optic  axis  is  parallel  to  the  surface  and  to  the 
plane  of  incidence  (Fig.  595).  The  two  disturbances  in  the  crystal  will 
travel  to  0  and  E  respectively  while  the  wave  travels  from  B  to  C  in  air. 
The  tangent  planes  CO  and  CE  are  the  two  wave  fronts.  The  disturbance 
at  E  is  due  to  A,  a  point  not  on  the  normal  to  the  extraordinary  wave  front 
passing  through  E.  The  wave  velocity,  or  the  velocity  of  the  wave  front, 
is  proportional  to  the  normal  distance  AN]  the  ray  velocity,  or  actual 
velocity  of  the  disturbance,  is  proportional  to  AE. 

When  the  axis  is  parallel  to  the  surface,  but  at  right  angles  to  the  plane  of 
incidence,  the  wave  front  is  found  as  shown  in  Fig.  596.  In  this  case,  the 
extraordinary  wave  also  has  a  circular  section.  Only  in  this  plane  of 
incidence  is  the  ratio  (sin  t)/(sin  r)  constant  for  the  extraordinary  ray, 


DOUBLE   REFRACTION  AND  POLARIZATION 


657 


and  this  ratio  is  called  ne,  the  extraordinary  index  of  refraction.  The 
value  of  the  ratio  V/Ve  varies  with  the  direction  in  every  other  plane  of 
incidence,  and  hence  cannot  properly  be  called  the  index  of  refraction. 

The  general  case,  where  the  axis  is  at  any  angle  with  the  refracting  sur- 
face and  the  plane  of  incidence,  is  shown  in  Fig.  597. 


B 


FIG.  596. 


Fio.  597. 


Fia.  598. 


762.  Uniaxial  Prisms. — When  light  is  incident  on  a  doubly-refracting 
prism  with  its  axis  parallel  to  the  refracting  edge  (Fig.  598),  the  ordinary 
and  the  extraordinary  rays  will  be  separated,  and  the  angular  divergence 
will  persist  after  emergence.  Two  spectra  will  be  formed,  with  light 
polarized  in  opposite  planes.  The  ordinary  spectrum  will  be  less  deviated 
than  the  extraordinary  by  a  quartz  prism  and  more  by  a  calcite  prism. 
When  the  optic  axis  is  parallel  to  the  refracting  edge  of  the  prism  the  two 
indices  of  refraction  may  be  determined  from  the  re'ations 

sin  $  (A  +  DO)       ,          sin  £  (A  +  Z>«) 
n0  =»  — -; — j— j —  ana  ne  =  —    •    \  A — 

sin  5  A  sin  £  A 

Some  values  of  the  indices  of  refraction  for  sodium  light  are  given  below: 

Positive  Crystals:  n0  ne 

Quartz 1.5442         1.5533 

Ice 1.3091         1.3104 

Negative  Crystals: 

Calcite  (Iceland  spar) 1 . 6584         1 . 4864 

Beryll 1.5740         1.5674 

Sodium  nitrate 1.5874         1.5361 

The  difference  between  n0  and  ne  is  greater  in  the  case  of  Iceland  spar 
than  in  any  other  ordinary  crystal. 

42 


658 


LIGHT 


763.  Polarizing  Prisms. — The  two  polarized  rays  produced  by 
a  doubly  refracting  crystal  are  not  sufficiently  separated  to  be 
conveniently  used  when  a  single  beam  is  desired.     The  separation 
may  be  increased  by  using  an  ordinary  triangular  prism,  but 

this  introduces  dispersion,  so  that  other  devices 
must  be  employed.  The  most  common  is  the 
rhombohedral  prism  invented  by  Nicol,  of  Edin- 
burgh, in  1828.  In  the  principal  section  of  a 
crystal  of  calcite  (Fig.  599)  the  angles  at  B  and 
D  are  71°.  The  two  end  faces  AB  and  CD  are 
cut  down  to  A'B  and  C'D,  so  that  these  angles 
are  reduced  to  68°.  The  crystal  is  then  sliced 
along  A'C'  in  a  plane  perpendicular  to  the  ends 
and  to  the  principal  section.  The  two  surfaces 
are  polished  and  cemented  together  with  Canada 
balsam,  which  has  an  index  of  refraction  less 
than  that  of  the  calcite  for  the  ordinary  and 
greater  for  the  extraordinary  ray.  If  a  ray  of 
light  r  is  incident  in  a  direction  parallel  to  the 
edge  AD  the  ordinary  ray  will  be  totally  reflected 
from  the  Canada  balsam,  while  the  greater  por- 
tion of  the  extraordinary  ray  will  be  transmitted.  The  reduction 
of  the  angles  at  A  and  D  is  for  the  purpose  of  securing  the  proper 
angle  of  incidence  on  the  balsam  to  produce  this  effect. 

The  Foucault  prism  resembles  that  of  Nicol,  but  the  total 
reflection  is  from  an  air  film.  This  allows  the  prism  to  be  made 
shorter,  but  there  is  a  greater  loss  of  light  by  reflection  and  a 
smaller  field  of  view. 

764.  The  Polariscope  is  an  instrument  for  the  study  of  the 
optical  properties  of  substances  with  respect  to  polarized  light. 
It  consists  of  two  Nicol  prisms  or  piles  of  plates,  one  called  the 
polarizer,  to  produce  the  polarized  light,  the  other,  the  analyzer, 
which  may  be  set  with  its  principal  section  at  any  desired  angle 
with  that  of  the  polarizer,  to  test  the  incident  light  with  respect 
to  the  nature  and  direction  of  its  polarization.     If  any    doubly 
refracting  substance  is  placed  between  the  two  its  effects  on  the 
polarized  light  transmitted  through  it  may  be  studied  by  the 
analyzer. 


Fio.  599. 


DOUBLE  REFRACTION  AND  POLARIZATION  659 

« 

765.  Resolution  and  Composition  of  Vibrations. — If  the  polar- 
izer and  analyzer  are  set  with  their  principal  sections  parallel, 
light  which  has  traversed  the  first  will  pass  through  the  second 
without  sensible  loss.  If  their  principal  sections  are  at  right 
angles  to  each  other,  or  "set  for  extinction,"  no 
light  w;ll  be  transmitted  through  the  analyzer.  If 
the  angle  between  the  principal  sections  is  a  (Fig. 
600),  and  if  a  is  the  amplitude  of  the  light  trans- 
mitted by  the  first  Nicol,  the  amplitude  of  that 
transmitted  through  the  second  is  a  cos  a,  and  its  / 
intensity  is  proportional  to  a2  cos2  a.  The  intensity  FIO.  eoo. 
of  the  totally  reflected  ordinary  ray  is  a2  sin3  a. 
The  sum  of  the  two  intensities  is  a2(cos2a-fsin2a)  =a2,  which  is 
equal  to  the  intensity  of  the  light  incident  on  the  analyzer.  This 
simple  law  of  resolution  of  vibrations  into  components  by  double 
refraction,  giving  determinate  control  of  the  intensity  through  a 
wide  range,  is  made  use  of  in  several  forms  of  photometer. 

If  the  two  Nicols  are  replaced  by  two  crystals  of  calcite  with 
their  principal  sections  at  an  angle  of  a  with  each  other,  as  in 
Huyghens'  experiment  (§  752),  an  ordinary  ray  o  and  an  extra- 
ordinary ray  e  of  the  same  amplitude  a  are  produced  by  the 
resolution  of  the  vibrations  along  two  directions  in  the  first 
crystal.  At  incidence  on  the  second  crystal,  the  ordinary  ray 
will  be  resolved  into  the  components  oo'  and  oe'  of  amplitudes 
a  cos  a  and  a  sin  a  and  the  extraordinary  ray  into  the  components 
eof  and  ee',  of  amplitudes  a  sin  a  and  a  cos  a.  There  will  be, 
therefore,  in  general  four  rays,  as  found  by  Huyghens,  which 
will  be  of  equal  intensity  when  a =45°.  When  the  principal 
planes  are  at  right  angles,  the  incident  ordinary  ray  goes  through 
the  second  crystal  as  an  extraordinary  ray  and  the  extraordinary 
as  an  ordinary  ray,  and  there  are  only  two  images. 

If  the  second  crystal  is  replaced  by  a  Nicol  prism,  with  its 
principal  section  parallel  to  that  of  the  first  crystal,  only  the 
components  oe'  and  ee'  emerge,  their  vibrations  being  in  the 
same  plane,  that  of  the  principal  section  of  the  analyzer.  If 
the  two  rays  are  superimposed  on  emergence,  the  intensity  will 
depend  not  only  on  the  amplitudes  of  the  two  components,  but 
on  the  phase  differences  which  have  been  introduced  owing  to 
the  difference  in  velocity  in  the  crystal  of  the  two  rays  from 


660  .  LIGHT 

• 

which  these  components  are  derived;  in  other  words,  there  may 
be  interference  provided  the  light  falling  on  the  first  crystal  is 
plane  polarized  (see  next  section.) 

766.  Interference  of  Parallel  Polarized  Light. — If  parallel  plane 
polarized  white  light  passes  through  a  double  refracting  crystal  of 
uniform  thickness  t  and  then  through  an  analyzer,  uniform  colored 
effects  are  produced  over  the  entire  field,  since  some  colors  are 
reenforced  and  some  weakened  by  interference.     There  is  no  real 
loss  or  gain  for  any  color,  for,  as  shown  in  §765,  whatever  energy 
is  lost  in  the  extraordinary  ray  is  gained  by  the  ordinary,  and 
conversely,  so  that  the  ordinary  light  which  is  internally  reflected 
in  the  prism  is  complementary  to  that  passing  through.     When 
the  principal  sections  of  the  crystal  and  the  analyzer  are  either 
parallel  or  at  right  angles  to  each  other  no  modification  of  the 
light  is  produced,  the  ordinary  or  the  extraordinary  ray  alone 
getting  through,  so  that  there  can  be  no  interference.     In  all 
other  positions  of  the  analyzer  there  are  varying  proportions  of 
white  and  colored  light  transmitted,  the  color  effects  being  most 
pronounced  when  the  principal  sections  are  at  an  angle  of  45° 
with  each  other. 

The  original  beam  of  light  falling  on  the  crystal  must  be  plane 
polarized.  If  ordinary  light  is  used  the  succession  of  waves 
vibrating  in  different  planes  when  resolved  in  the  crystal  will 
give  rise  to  all  possible  distributions  of  amplitudes,  so  that  all 
colors  will  be  equally  affected  and  the  resultant  effect  will  be 
white  light. 

767.  Double  Refraction  due  to  Strain. — If  a  plate  of  glass  or 
other  isotropic  substance  is  placed  between  a  polarizer  and  an 
analyzer  set  for  extinction  no  effect  is  produced.     If  the  sub- 
stance is  then  compressed  or  stretched  some  light  will  pass  and 
interference  effects  similar  to  those  described  above  will  be 
produced.     This   shows   that   an  isotropic  substance  becomes 
doubly  refracting  when  subjected  to  unsymmetrical  strain.     This 
method  offers  a  very  delicate  test  of  deviations  from  isotropy. 
Some  liquids  show  the  same  characteristics  in  cases  where  the 
viscosity  is  so  great  or  the  stress  so  suddenly  applied  that  a 
uniform   hydrostatic   pressure   has   not   had   time   to    become 
established  throughout  the  substance.     Imperfectly  annealed 
glass  exhibits  double  refraction.     As  shown  by  Tyndall,  a  bar  of 


DOUBLE  REFRACTION  AND  POLARIZATION 


661 


glass  set  in  longitudinal  vibration  restores  the  light  through  the 
crossed  nicols,  and  a  rotating  mirror  shows  that  the  effect  is  set 
up  periodically  as  the  compression  waves  pass  across  the  field. 

Kerr  found  that  a  block  of  glass  in  a  strong  electrostatic  field 
becomes  doubly  refracting  like  a  uniaxial  crystal  with  its  axis 
parallel  to  the  field. 

768.  Interference  of  Convergent  or  Divergent  Light. — If  a  divergent  or 
convergent  pencil  of  polarized  light  falls  on  a  doubly  refracting  crystal 


Fio.  601 

different  portions  of  the  pencil  will  traverse  the  crystal  at  different  angles, 
and  therefore  with  different  optical  paths,  hence  the  interference  effects  will 
not  be  uniform  over  the  whole  field.  In  general  the  effects  are  quite  com- 
plex and  cannot  be  discussed  here,  but  the  simple  case  of  a  uniaxial  crystal 
cut  perpendicularly  to  the  optic  axis  may  be  considered  as  an  illustration. 
Consider  such  a  pencil  diverging  from 
S  and  falling  normally  on  the  face 
ABCD  of  a  doubly  refracting  crystal 
with  its  axis  parallel  to  AAf  (Fig.  601). 
The  vibrations  of  the  incident  light  may 
be  supposed  to  be  in  a  vertical  plane,  as 
indicated  by  the  arrows.  At  P  and  Q 
the  incident  vibrations  are  respectively 
parallel  and  perpendicular  to  the  prin- 
cipal sections  PPf  and  QQ'  and  travel 
through  without  change.  If  an  analyzer 
is  placed  beyond  the  crystal  and  set  for 
transmission  or  extinction  of  light  trans- 
mitted by  the  [polarizer  there  will  be  a 
light  cross  or  a  dark  cross  on  a  screen 

beyond  it  corresponding  to  the  crossed  lines  PP'  and  QQ'.  The  light 
incident  at  such  a  point  as  R,  however,  will  be  vibrating  at  an  angle  with 
the  principal  section  RR',  and  will  be  resolved  into  two  components.  A 
relative  difference  of  phase  between  them  will  exist  at  emergence,  and 
interference  effects  will  take  place  when  they  are  re-resolved  into  the  same 
plane  by  the  analyzer.  The  same  difference  of  path  will  exist  for  all  rays 


Fio    602. 


662  LIGHT 

incident  at  the  same  angle,  that  is,  at  all  points  equidistant  from  the 
normal  from  S  to  ABCD,  hence  colored  rings  similar  to  Newton's  rings  in 
appearance  will  be  projected  on  a  screen  beyond  the  analyzer.  The  "  rings 
and  brushes"  due  to  a  calcite  plate  are  shown  in  Fig.  602.  The  brushes 
are  dark,  showing  that  the  Nicols  are  crossed. 

The  interference  effects  due  to  crystals  cut  in  other  ways  or  to  biaxial 
crystals  are  analogous  to  those  described  above,  but  more  complex. 

769.  Circular    and   Elliptical   Polarization. — Consider  the  state  of  the 
light  originally  plane  polarized  as  it  emerges  from  a  doubly  refracting 
crystal  before  it  reaches  the  analyzer.     The  ordinary  and  extraordinary 
rays  start  from  the  first  surface  in  the  same  phase,  but,  as  their  velocities 
are  different,  one  set  of  waves  will  fall  behind  the  other.     At  different 
points  within  the  crystal  there  will  be  two  disturbances  at  right  angles 
to  each  other  and  with  phase  differences  depending  upon  the  thickness 
of  the  medium  traversed.     The  optical  difference  of  path  d  at  a  distance  / 
from  the  first  surface  is  [(VI  Fe)  -  (V/  Vo)]t.     At  points  where  this  difference 
is  nX  the  light  is  plane  polarized  in  a  direction  intermediate  between  the 
planes  of  vibration  of  the  two  components,  the  slope  depending  on  their 
relative  amplitudes,  and  being  45°  if  these  are  equal.     If  the  difference 
of  path  is  (2n  + 1)/2.  A  the  light  will  likewise  be  plane  polarized,  but  with  a 
reversed  direction  of  slope.     If  the  difference  of  path  is  any  odd  multiple 

of  a  quarter  of  a  wave-length  the 
disturbance   will  be  elliptical,   or 

I       0      O        Q  \        ^         circular  if  the  amplitudes  are  equal. 

!  ;         For    intermediate    differences    of 

"• • • ! ! —     path  the  disturbance  will  be  ellip- 

F,o  «of       '^     %X      «<*!.  the  axes  of  the  ellipse  being 
oblique  with  respect  to  the  axis 

of  the  crystal.  The  successive  stages  at  different  distances  from  the  first 
surface  are  shown  in  Fig.  603.  On  emergence  from  the  crystal  the 
disturbance  will  preserve  the  final  form,  and  will  be  plane,  elliptically,  or 
circularly  polarized  according  to  the  thickness  of  the  crystal.  If  the  waves 
are  circularly  polarized  the  disturbance  travels  through  space  like  a  point  on 
a  rotating  screw.  The  polarization  is  said  to  be  right-handed  if  the  rotation 
is  clockwise  looking  in  the  direction  of  propagation,  or  if  the  displacement 
resembles  that  of  a  right-handed  screw,  left-handed  if  the  displacement  is 
like  that  of  a  left-handed  screw. 

When  light  is  totally  reflected  there  is  a  phase  difference  between  the 
vibrations  respectively  in  and  at  right  angles  to  the  plane  of  incidence,  so 
that  this  light  is  elliptically  or  circularly  polarized.  In  ordinary  reflection 
there  is  a  slight  elliptical  polarization,  which  becomes  very  marked  in  the 
case  of  metallic  reflection. 

770.  Production  and  Detection  of  Elliptically  Polarized  Light. — Circularly 
or  elliptically  polarized  light  cannot  be  detected  by  the  unaided  eye.     If 
viewed  through  a  Nicol  prism  no  change  in  the  intensity  of  circularly  polar- 
ized light  accompanies  rotation  of  the  prism,  as  a  component  of  unchanging 
magnitude  is  transmitted.     If  the  light  is  ellipitcally  polarized  there  will  be 


DOUBLE  REFRACTION  AND  POLARIZATION 


663 


FIG.  604. 


variations  of  intensity  as  the  prism  is  rotated,  the  intensity  being  greatest 

when  the  principal  section  of  the  prism  is  parallel  to  the  major  axis  of  the 

ellipse  (component  amplitude  of  greatest  magnitude)  and  a  minimum  when 

it  is  parallel  to  the  minor  axis.     If  circularly  polarized  light 

passes  through  a  crystal  producing  a  relative  retardation  of 

an  odd  number  of  quarter  wave-lengths  of  a  particular  color 

the  additional  retardation  between  the  components  will 

cause  the  emergent  light  to  be  plane  polarized  hi  an  azimuth 

which  may  be  found  by  the  analyzing  Nicol  prism.     Such 

a  crystal  is  called  a  quarter-wave  plate.     These  plates  can 

readily  be  prepared  from  thin  sheets  of  mica. 

Another  device  for  securing  or  testing  circularly  polarized 
light  is  Fresnel's  rhomb  (Fig.  604).  A  block  of  glass  is  cut 
with  the  angle  at  A  equal  to  54°,  so  that  a  pencil  of  light 
incident  normally  will  be  totally  reflected  at  B  and  again  at 
C,  the  angle  of  incidence  Toeing  54°.  At  each  reflection  at 
this  particular  angle  a  phase  difference  of  an  eighth  of  a 
period  is  introduced  between  the  vibrations  in  and  at  right 
angles  to  the  plane  of  incidence,  and  the  emergent  light  is 
circularly  polarized  if  the  incident  light  is  plane  polarized  at 
an  angle  of  45°  with  the  plane  of  incidence.  If  this  angle 
differs  from  45°  the  amplitude  of  the  two  components  will  be  different  and 
the  light  will  be  elliptically  polarized. 

771.  Rotation  of  the  Plane  of  Polarization. — If  two  Nicol  prisms 
are  set  for  extinction  and  a  crystal  of  quartz  cut  with  the  face  on 
which  the  light  falls  at  right  angles  to  the  axis,  or  a  solution  of 
sugar  or  tartaric  acid,  is  placed  between  them,  the  light  will  be 
restored.  On  turning  the  analyzer  through  a  given  angle 
depending  on  the  thickness  of  the  crystal  or  the  solution,  the 
light  will  again  be  extinguished.  This  shows  that  the  plane  of 
polarization  has  been  rotated  through  this  angle.  Substances 
producing  this  effect  are  said  to  be  naturally  optically  active. 

Some  quartz  crystals  rotate  the  plane  of  polarization  clock- 
wise looking  in  the  direction  of  propagation,  and  are  called  right- 
handed;  others  produce  rotation  in  the  opposite  direction,  and  are 
called  left-handed.  These  two  classes  of  crystals  can  be  distin- 
guished by  inspection  on  account  of  certain  unsymmetrical 
facets  which  are  differently  placed  in  the  two  cases. 

The  rotation  of  the  plane  of  polarization  of  light  of  the  wave 
lengths  corresponding  to  some  Fraunhofer  lines  caused  by  a 
quartz  plate  of  one  mm.  thickness  is  given  below: 

A  B  C  D  F  G  K 

12.67°     15.75°     17.32°     21.70°     32.97°     42.60°     52.15° 


664 


LIGHT 


As  shown  by  these  figures,  the  rotation  varies  very  nearly  in- 
versely as  the  square  of  the  wave-length. 

Fused  quartz  shows  no  double  refraction  or  rotation.  These 
effects  are  evidently  due  rather  to  the  crystalline  arrangement 
of  the  molecules  than  to  their  individual  structure. 

If  light  passes  through  a  quartz  prism  so  cut  that  the  light  is 
transmitted  in  the  direction  of  the  optic  axis  it  is  found  that  there 
is  a  slight  double  refraction,  so  that  spectral  lines  appear  double. 
This  shows  that  the  two  waves  travel  with  slightly  different 
velocities  even  along  the  optic  axis;  consequently  the  two  wave 
surfaces  cannot  be  tangent  to  each  other  (§754),  but  must  be 
slightly  separated.  This  is  not  generally  true  of  uniaxial  crystals, 
but  only  of  those  which  rotate  the  plane  of  polarization.  It  is 
found  that  the  two  waves  are  circularly  polarized  in  opposite 
directions,  so  that  this  is  a  case  of  circular  double  refraction. 
As  first  suggested  by  Fresnel,  it  appears  that  when  light  travels 
along  the  optic  axis  of  quartz  it  is  divided  into  two  circularly 
polarized  components,  which  travel  with  different  velocities. 
These  on  emergence  recombine  to  form  plane  polarized  light 
but  in  a  different  plane.  This  offers  a  simple  explanation  of  the 
rotation. 

If  each  circular  displacement  r  and  I  is  resolved  into  two  linear  displace- 
ments x  and  y,  it  is  seen  that  when  the  two  velocities  of  propagation  of  the 
two  circular  components  are  equal  (Fig.  605)  the  two  x  components  at  any 
point  in  the  medium  are  equal  and  opposite,  leaving  the  two  y  components 


A",       A 


in  the  same  direction  to  combine  in  a  plane  polarized  beam,  the  vibrations  of 
which  are  in  the  same  direction  as  those  of  the  original  beam.  But  when  the 
velocities  of  propagation  are  unequal  (Fig.  606)  the  x  components  and  the 
y  components  are  respectively  unequal.  If,  however,  we  refer  displacements 


DOUBLE  REFRACTION  AND  POLARIZATION  665 

to  an  axis  of  reference  shifted  through  an  angle  (a,—  aa)/2  with  the  original 
direction  of  vibration  it  will  be  seen  that  with  reference  to  this  axis  the  x 
displacements  will  cancel  each  other.  This  line  A'B'  then  represents  the 
final  direction  of  vibration  and  the  rotation  is  (at  —  «2)/2. 

772.  .Rotation  by  Liquids.  Saccharimetry. — A  number  of 
liquids,  such  as  turpentine  and  the  different  sugars  in  solution, 
also  cause  rotation,  that  due  to  turpentine  being  left-handed, 
and  that  due  to  some  sugars  right-handed,  of  others  left-handed. 
The  vapors  of  such  substances  as  turpentine  also  produce  rota- 
tion. In  such  cases,  as  with  quartz,  there  is  circular  double 
refraction,  and  the  rotation  is  to  be  explained  in  the  same  way. 
In  the  case  of  liquids  and  vapors,  however,  the  effect  must  be  due 
to  unsymmetrical  structure  of  the  molecule  itself,  as  there  is  no 
crystalline  structure,  or  if  there  is,  the  crystals  are  irregularly 
oriented.  The  amount  of  rotation  varies  inversely  as  the  square 
of  the  wave-length,  and  is  proportional  to  the  thickness  of  the 
medium,  and  also  to  the  concentration  in  the  case  of  solutions. 

Hence  if  a  is  the  rotation  of  light  of  a  definite  wave-length  in 
passing  a  distance  of  I  (decimeters)  through  a  substance  of  density 
p  gm/cm3  or  of  percentage  concentration  p,  a  =  [a]Z<o  =  [a]Zp/100, 
where  [a]  is  a  constant  for  the  substance  called  its  specific  rotatory 
power. 

The  rotatory  power  of  the  sugars  is  slightly  affected  by  the 
presence  of  impurities.  The  percentage  of  sugar  may  be  deter- 
mined by  measuring  the  rotation  with  a  sensitive  polariscope. 
This  is  called  saccharimetry.  Most  sugars  rotate  to  the  right,  but 
levulose  rotates  to  the  left.  In  some  cases  the  specific  rotatory 
power  varies  slightly  with  the  concentration,  and  that  of  levulose 
is  influenced  by  the  temperature.  The  specific  rotatory  power 
for  sodium  light  for  some  sugars  at,  20°G.  is  given  below  (from 
Landolt,  Optical  Rotation).  The  positive  sign  indicates  right- 
handed,  the  negative  left-handed  rotation,  while  p  is  the  concen- 
tration. 

Sucrose  (cane  sugar)  -f  66 . 44°  +  0 . 0087p 

Dextrose  +  52. 50° +0.0188? 

Levulose  -   88.13°-0.2583p 

Lactose  (milk  sugar)  +  52.53° 

Maltose  (malt  sugar)  + 140 . 4  °  -  0 . 0184p 


666  LIGHT 

773.  Rotation  by  Magnetic  Field. — Faraday  discovered  that  the  plane  of 
polarization  of  light  passing  through  a  refractive  substance  in  a  magnetic 
field  is  rotated  if  the  light  travels  parallel  to  the  force  lines.     No  effect  is 
produced  by  a  magnetic  field  on  light  waves  in  free  space,  and  in  general 
the  effect  increases  with  the  refractive  power  of  the  substance,  being  espe- 
cially marked  in  dense  flint  glass  and  carbon  bisulphide  and  very  feeble  in  the 
case  of  gases.     The  rotation  is  usually  proportional  to  the  field  intensity  and 
to  the  thickness  of  the  medium.     Some  substances  cause  right-handed  and 
others   left-handed   rotation.     The    effect   varies   with    the   wave-length. 
The  rotation  produced  by  1  cm.  thickness  in  a  field  of  unit  strength  (Verdet's 
constant)  is:  For  water,  0.0131°;  carbon  bisulphide,  0.0435°;  dense  flint 
glass,  0.06°.     Enormous  rotations  are  produced  by  thin  films  of  iron  or 
other  magnetic  material  in  a  strong  magnetic  field. 

In  naturally  active  substances  the  direction  of  rotation  is  independent  of 
the  direction  of  propagation  of  the  light,  so  that  if  a  rotated  beam  is  reflected 
its  plane  is  turned  back  to  the  original  position.  In  magnetically  active 
substances  the  direction  of  rotation  is  reversed  with  reversal  of  the  field,  so 
that  if  the  beam  is  reflected  through  the  medium  the  rotation  is  doubled. 

774.  Kerr  Effect. — When  a  beam  of  plane-polarized  light  is  reflected  from 
a  metallic  surface  a  relative  phase  difference  is  introduced  between  com- 
ponents respectively  in  and  at  right  angles  to  the  plane  of  incidence,  so 
that  the  reflected  light  is  elliptically  polarized,  unless  the  incident  light 
vibrates  parallel  or  at  right  angles  to  the  plane  of  incidence.     Kerr  found 
that  if  the  light  is  reflected  from  the  polished  pole  of  an  electromagnet  it 
becomes  slightly  elliptically   polarized,   even   under   the   conditions   just 
mentioned. 

776.  Zeeman  Effect. — Zeeman  placed  a  bunsen  flame  colored  with  sodium 
between  the  poles  of  a  powerful  electromagnet.  When  the  light  from  the 
source  traveled  either  parallel  or  at  right  angles  to  the  direction  of  the  field, 
he  observed  a  broadening  of  the  spectral  lines  when  the  field  was  established. 
H.  A.  Lorentz  pointed  out  that  such  effects  were  in  harmony  with  the 
electron  theory  of  radiation  proposed  by  him,  and  predicted  that  further 
investigation  would  show  the  radiation  to  be  polarized  by  the  field,  either 
circularly  or  plane,  according  to  the  direction  in  which  it  was  viewed. 
Zeeman  found  this  to  be  the  case.  In  the  simplest  cases,  when  the  light  is 
viewed  normally  to  the  field,  each  spectral  line  is  split  into  triplets,  the 
vibrations  in  the  central  and  undisplaced  component  being  parallel  to  the 
force  lines,  those  of  the  lateral  and  displaced  components  at  right  angles  to 
the  force  lines.  When  the  source  is  viewed  parallel  to  the  force  lines  single 
lines  become  doublets,  the  components  being  circularly  polarized  in  opposite 
directions,  and  displaced  on  each  side  of  the  mean  position  of  the  line.  In 
some  cases  the  effects  are  much  more  complex,  a  large  number  of  compo- 
nents being  produced  from  single  lines,  but  the  simple  case  described  above 
is  fully  explained  by  Lorentz's  theory  which  assumes  that  the  light  waves 
are  disturbances  caused  by  rotations  of  these  electrons  about  the  atoms  of 
the  source,  and  that  the  motion  of  the  electrons  is  modified  by  the  magnetic 
field. 


DISPERSION  AND  SELECTIVE  REFLECTION  667 

DISPERSION  AND  SELECTIVE  REFLECTION 

776.  Dispersion. — It  was  pointed  out  in  §675  that  dispersion  due  to 
refraction  is  irrational,  that  is,  there  is  no  simple  relation  between  the 
deviation  of  lines  in  the  spectrum  produced  by  a  prism  of  the  substance  and 
the  wave-lengths,  as  there  is  in  diffraction  spectra.  As  a  general  rule  the 


1.51 
1.50 


1.45 


1.40 


1.35 


1.31 


.25.50.75.100 


2/* 


3fJ.         4/Jt.         5/A 
Fio.  607. 


longer  waves  are  less  refracted  than  the  shorter,  and  the  dispersion  steadily 
diminishes  in  the  direction  of  the  longer  waves,  so  that  the  red  end  of  the 
spectrum  is  "telescoped"  as  compared  with  the  violet.  Within  the  limits 
of  the  visible  spectrum  the  relation  between  the  index  of  refraction  and 
the  wave-length  is  closely  expressed  by  the  empirical  relation  (Cauchy's 
formula) 


where  A,  B,  and  C  are  constants  varying  with  the  substance.  The  disper- 
sion curve  of  fluorite,  showing  the  relation  between  index  of  refraction  and 
wave-length,  is  shown  in  Fig.  607. 

777.  Anomalous  Dispersion. — It  is  not  always  true  that  the  deviation  of 
waves  by  refraction   increases  as  the   waves 
become  shorter.     Iodine  vapor  transmits  only 
the  red  and  violet,  and  the  red  is  refracted  more 
than  the  violet.     In  the  case  of  fuchsine,  an  ani- 
line dye,  blue  and  violet  are  less  refracted  than 
red,  the  green  is  absorbed,  and  the  other  colors 
occur  in  the  usual  order.     Such  anomalous  dis- 
persion is  shown  not  only  by  a  large  number  of  FIO.  gog. 
substances  such  as  the  aniline  dyes,  but  by  the 

vapors  of  sodium  and  other  metals,  and,  in  fact,  by  almost  every  substance 
investigated  in  some  part  of  its  spectrum,  visible  or  invisible.  Anomalous 


668  LIGHT 

dispersion  always  occurs  in  the  neighborhood  of  what  appears  to  be  a  strong 
absorption  band,  which  is,  more  properly,  a  region  where  the  light  is  selec- 
tively reflected  rather  than  transmitted  or  absorbed.  The  index  of  refrac- 
tion is  abnormally  increased  on  one  side  of  this  band  and  diminished  on  the 
other,  resulting  in  the  reversal  of  the  corresponding  colors  in  a  spectrum 
formed  by  a  prism  of  the  substance.  The  dispersion  curve  of  a  substance 
between  two  regions  of  such  selective  reflection  is  shown  in  Fig.  608. 
Between  these  regions  the  curve  resembles  the  normal  dispersion  curve 
shown  in  Fig.  607. 


Fio.  609. 

The  absorptive  power  of  substances  showing  anomalous  dispersion  is 
usually  so  great  that  it  is  impossible  to  secure  a  prism  of  sufficient  angle  to 
give  a  spectrum  long  enough  to  clearly  show  the  effect.  The  method  of 
crossed  prisms  is  well  adapted  for  showing  it.  If  light  passes  in  succession 
through  two  prisms  with  their  refracting  edges  at  right  angles  to  each  other 
the  resultant  spectrum  will  usually  be  a  line  or  smooth  curve  inclined  in 
direction  to  the  edges  of  both  prisms.  If,  however,  one  of  the  prisms  gives 
anomalous  dispersion  the  resultant  spectrum  will  be  broken  and  irregular, 
as  shown  in  Fig.  609,  which  illustrates  the  anomalous  dispersion  of  sodium 
vapor  in  the  neighborhood  of  the  D  lines. 

778.  Selective  Reflection. — The  color  of  natural  objects  is  primarily  due 
to  selective  absorption,  the  effective  waves  being  those  which  escape  absorp- 
tion and  become  scattered.  In  the  case  of  substances  showing  surface  color, 
however,  the  effect  is  due  to  selective  reflection,  and  the  transmitted  light  is 
complementary  to  that  reflected.  This  is  the  case  with  substances  showing 
anomalous  dispersion.  The  colors  which  are  selectively  reflected  lie  between 
the  colors  which  are  transmitted  and  anomalously  dispersed.  The  so-called 
"absorption"  bands  so  often  referred  to  in  this  connection  are  thus  seen  to 
be  largely  due  to  lack  of  transmission  because  of  reflection.  As  the  reflect- 
ing power  of  the  substance  is  abnormally  high  in  such  regions,  they  are  said 
to  show  metallic  reflection  for  the  colors  concerned.  Recent  investigations 
show  that  most  substances  exhibit  anomalous  dispersion  in  some  region  of 
their  spectrum.  For  example,  quartz,  rock  salt,  and  fluorite  show  anoma- 
lous dispersion  and  metallic  reflection  for  certain  very  long  waves.  For 
radiation  of  wave-length  611,000  Angstrfim  units  (reflected  from  sylvite) 


DISPERSION  AND  SELECTIVE  REFLECTION  669 

quartz  has  an  index  of  refraction  of  2.12,  considerably  greater  than  that  of 
the  shortest  ultra-violet  waves. 

779.  Theory  of  Anomalous  Dispersion  and  Selective  Reflection. — It  is 
believed  that  these  effects  are  due  to  resonance,  the  free  periods  of  the 
vibrating  parts  of  the  molecules  being  the  same  as  that  of  the  waves  selectively 
reflected.  The  vibrating  element  of  the  molecule  is  probably  the  elec- 
tron. Selective  reflection  may  be  considered  as  the  re-radiation  of  ether 
waves  by  the  electrons,  just  as  a  tuning  fork  re-radiates  sound  waves  after 
being  excited  by  resonance.  There  is  in  such  cases  little  "f notional" 
absorption  of  energy,  which  is  completely  transformed,  to  heat,  not  re-radi- 
ated. It  may  be  shown  from  mechanical  analogies  and  electrical  theory 
that  the  rate  of  propagation  of  waves  through  a  medium  will  be  accelerated 
or  retarded  if  the  medium  contains  vibrating  elements  which  have  a  free 
rate  of  vibration  slightly  greater  or  less  than  that  of  the  waves. 

A  complete  dispersion  formula,  taking  account  of  regions  having  anoma- 
lous dispersion  for  wave-lengths  ^  and  A,  is 

n'-A-f      B  ° 

+  p^2+^T^ 

where  Jt  and  X2  are  the  lengths  of  the  light  waves  having  the  same  rate 
of  vibration  as  the  electrons  of  the  substance.  This  gives  a  discontinuity 
in  n,  the  refractive  index,  and  anomalous  dispersion  for  these  wave-lengths. 
The  electron  theory,  first  put  on  a  definite  basis  by  Zeeman's  discovery, 
suggests  an  explanation  of  radiation  and  most  of  the  optical  properties  of 
bodies, 

References 

PRESTON'S  Theory  of  Light. 

WOOD'S  Physical  Optics. 

EDSER'S  Light  for  Students. 
These  three  books  give  a  more  advanced  treatment  of  the  subject  than 

the  ordinary  text-book,  but  contain  much  interesting  material  which  can  be 

understood  by  beginners. 

TAIT'S  Light  also  gives  a  somewhat  advanced  treatment 

TYNDALL'S  Light. 

STOKES'  Lectures  on  Light. 

HASTING'S  Light. 

These  three  books  give  a  very  interesting  and  simple  popular  account  of 

the  subject. 

S.  P.  THOMPSON'S  Light,  Visible  and  Invisible,  is  an  exceedingly  interesting 
popular  discussion  of  modern  discoveries,  not  only  in  light,  but  in 
related  phenomena,  such  as  Rontgen  rays  and  electric  waves. 

LE  CONTE'S  Sight  treats  of  the  eye  and  vision  in  a  popular  way. 

ABNEY'S  Color  Measurement  and  Mixture  and  Color  Vision  give  an  elemen- 
tary but  complete  discussion  of  the  subject. 

CHURCH'S  Colour  is  another  interesting  little  book  on  the  same  subject. 

BALY'S  Spectroscopy  is  an  excellent  presentation  of  spectroscopic  methods 
and  theory. 


670  LIGHT 

WATT'S  Introduction  to  Spectrum  Analysis  is  a  somewhat  more  popular 
book  than  the  above,  and  gives  tables  of  wave-lengths. 

CLERKE'S  Problems  in  Astrophysics  is  a  very  interesting  account  of  the  uses 
of  the  spectroscope  in  astronomical  work. 

MICHELSON'S  Light  Waves  and  their  Use  gives  an  account  of  interferometers 
and  their  applications. 

DERR'S  Photography  is  an  excellent  elementary  discussion  of  the  photo- 
graphic art. 

The  Scientific  Memoirs  Series  contains  the  following  reprints  of  important 
original  papers,  most  of  them  presented  in  simple  language:  Prismatic 
and  Diffraction  Spectra,  Fraunhofer;  The  Wave  Theory  of  Light, 
Huyghens,  Fresnel;  Laws  of  Radiation  and  Absorption,  Kirchhoff  and 
Bunsen;  The  Effects  of  a  Magnetic  Field  on  Radiation,  Faraday,  Kerr, 
Zeeman. 


Problems 

1.  A  man  is  5  feet  10  inches  high.     What  is  the  shortest   vertical  plane 

mirror  in  which  he  can  see  nis  full-length  image?  Ans.  35  in. 

T>  a    x.  2.  Two  plane  mirrors  are  parallel  to  each  other  at  a 

distance  of  30  cm.     Find  the  distance  from  each  mirror 

of  the  three  nearest  images  in  each  of  an  object  between  them  and  10 

cm.   from  one.  Ans.  10,  50,  70;  20,  40,  80. 

3.  A  beam  of  light  is  reflected  from  a  plane  mirror  revolving  clockwise 

about  a  vertical  axis  ten  times  per  second,  falls  on  a  neighboring  mirror 

revolving  anticlockwise  fifteen  times  per  second,   and  then  on  a  wall 

10  meters  away.     What  speed  does  the  spot  of  light  cross  the  wall? 

Ans.  3141.6m/sec.  anticlockwise. 

A  meter  rod  lies  along  the  axis  of  a  concave  mirror  of  20  cm.  focal 
length,  one  end  in  contact  with  the  mirror.  Describe  the  images  formed, 
and  calculate  the  position  of  the  first,  fifth,  tenth,  twentieth,  fortieth, 
and  one-hundredth  cm.  marks,  and  the  length  of  each  division  at  these 
points  (assuming  the  rod  to  be  2  cm.  wide) . 

Ans.  Virtual  distances,  1.05,  6.67,  20,   oo  ;  real,  40,  25.     Lengths,  2.1 

2.67,  4,  oo  ;  2,  0.5. 

5.  Prove  by  graphical  construction  the  statements  made  in  §666  concern- 
ing ellipsoidal,  hyperboloidal,  and  paraboloidal  mirrors. 

6.  Show  by  diagrams  the  successive  shapes  of  the  wave  reflected  from  a 
hemispherical  concave  mirror  as  it  passes  from  the  mirror  to  a  point 
beyond  the  focal  cusp.     (This  surface  must  everywhere  be  normal  to 
the  "rays"  which  it  cuts.), 

7.  A  convex  mirror  has  a  focal  length  of  25  cm.     Calculate  the  position 
and  the  height  of  the  image  of  an  object  10  cm.  high  and  15  cm.  in  front 
of  the  mirror.  Ans.  —  9.4,  6.3. 

8.  A  paper  square  with  sides  two  cm.  in  length  lies  in  and  parallel  to  the 
axis  of  the  above  mirror  at  a  distance  of  40  cm.     Describe  the  shape  oi 


PROBLEMS  671 

the  image,  and  calculate  the  lengths  of  its  sides  and  the  angles  between 

them.  Ana.  Quadrilateral;  sides  normal  to  axis,  0.765,  0.744. 

Distance  between  them,  0.364;  91°  39',  88°  21' 

9.  The  sun  has  an  angular  magnitude  of  32'.     What  is  the  size  of  the  solar 
image  formed  by  a  concave  mirror  of  50  ft.  focal  length?  Ans.  5.58  in. 
T»  £      j.  10.  A  layer  of  ether  (n  =  1.36)  2  cm.  deep  floats  on  water 

(n  =  1.33)  3  cm.  deep.     What  is  the  apparent  dis- 
tance of  the  bottom  of  the  vessel  below  the  surface?  Ans.  3.72. 
11.  An  object  is  viewed  through  a  cube  of  glass  (n  =  1.55)  10  cm.  thick,  in  a 
direction  at  an  angle  of  60°  with  the  normal  to  the  glass  surface.     What 
is  the  lateral  displacement  of  the  image?                               Ans.  5.29  cm. 
^                    12.  A  convex  lens  25  cm.  from  a  candle-flame  5  cm.  high 
forms  an  image  of  the  latter  on  a  screen.     When  the 
lens  is  moved  25  cm.  further  from  the  candle  an  image  is  again  formed 
on  the  screen.    Calculate  the  focal  length  of  the  lens ,  the  distance  of 
the  screen  from  the  candle,  and  the  size  of  the  two  images. 

Ans.  16.67;  75;  10,  2.5. 

13.  Show  by  graphical  construction  whether  it  is  possible  to  construct  a 
single  thick  double  convex  lens  which  will  give  a  real  erect  image;  and 
another  which  will  give  an  inverted  virtual  image. 

14.  A  candle  flame  100  cm.  from  a  convex  lens  of  focal  length  90  cm.  is 
displaced  2  cm.  away  from  the  lens  at  the  rate  of  1  cm.  per  second. 
What  is  the  displacement  and  the  average  velocity  of  its  image? 

cm. 

Ans.  135  cm.  toward  lens;  67.5 • 

sec. 

15.  A  convex  lens  (n  =  1.54)  has  a  focal  length  of  40  cm.  in  air.     What  is 
the  focal  length  in  water  (n  — 1.33)?  Ans.  136.8. 

16.  The  images  of  objects  seen  through  a  spherical  flask  or  cylindrical  glass 
of  uniform  thickness  are  of  diminished  size.     Explain. 

17.  Two  convex  lenses  of  focal  lengths  20  and  30  cm.  are  10  cm.  apart. 
Calculate  the  position  and  length  of  the  image  of  an  object  2  cm.  long 
100  cm.  in  front  of  the  first  lens.     (Consider  the  image  due  to  the  first 
lens  to  be  the  object  for  the  second.) 

Ana.  10  cm.  beyond  second  lens;  length  0.33. 

18.  Replace  the  first  lens  in  the  above  problem  by  a  concave  lens  of  the  same 
focal  length  and  determine  the  position  and  magnitude  of  the  image. 

Ans.  243  cm.  to  left  of  second  lens;  3.04. 

19.  When  focused  on  a  star,  the  distance  of  the  eye-piece  of  a  telescope  from 
the  object  lens  is  50  cm.     To  see  a  certain  terrestrial  object  clearly  the 
eye-piece  must  be  drawn  out  0.2  cm.     What  is  the  distance  of  the  object 
from  the  observer.  Ans.  125.5  m. 

20.  In  the  above  example,  if  the  eye-piece  has  a  focal  length  of  1  cm.,  and  if 
the  object  referred  to  is  a  tree  10  feet  high,  what  is  the  size  of  the  image 
formed  by  the  object  lens?    What  is  the  angular  magnitude  of  the 
image  formed  by  the  eye-piece?  Ans.  1.21  cm.;  62°  21'. 

21.  A  double  convex  lens  with  faces  having  a  radius  of  curvature  of  30  cm. 


672  LIGHT 

gives  a  real  image  at  a  distance  of  60  cm.  of  an  object  40  cm   away. 

What  is  its  focal  length?     Its  index  of  refraction?  Ans.  24 ;  1.625. 

22.  An  achromatic  lens  is  to  be  made  of  a  combination  of  a  crown  glass 

double  convex  lens  (n^  — 1.51,  np  ==1.52)  and  a  plano-concave  flint 

glass  lens  (HD  =  1.64,  np  «=1.66),  the  adjacent  faces  to  fit  together  and 

the  focal  length  to  be  50  cm.     Calculate  the  radii  of  curvature  of  the 

faces.  Ans.  r1  =  r2=»  —  ra>=19;  r4=oo  . 

Ph  trv       ^*  ^  can(^e  *B  Plftced  10  cm.  in  front  of  a  concave  mirror 

of  20  cm.  focal  length   (assumed  to  be  a  perfect 

reflector) .     What  is  the  illumination  on  a  screen  100  cm.  from  the  candle 

along  the  mirror  axis,  as  compared  with  7,  that  due  to  the  candle 

alone?  Ans.  3.31  7. 

24.  Solve  the  above  problem  after  substituting  a  convex  mirror  of  the  same 
focal  length  for  the  concave  mirror.  Ans.  1.33  7. 

25.  Two  sources  have  candle  power  16  and  97  respectively.     At  what  point 
between  them  must  a  screen  be  placed  to  be  equally  illuminated  by 
the  two?  Ans.  0.288  d  from  fainter  source. 

26.  Foucault  in  his  arrangement  for  measuring  the  velocity  of  light   (§644) 
placed  the  lens  between  the  source  and  the  revolving  mirror.     Show 
that  with  this  arrangement — (a)  The  stationary  mirror  must  be  concave, 
with  center  of  curvature  at  the  axis  of  the  revolving  mirror;  (6)  that  the 
stationary  mirror  cannot  be  placed  far  away  from  the  revolving  mirror 
unless  its  aperture  is  correspondingly  enlarged,  if  the  reflected  beam  is 
to    have   sufficient   intensity.     Show   that    Michelson's    arrangement 
obviates  these  disadvantages. 

n.         .  27.  A  60°  prism  has  an  index  of  refraction  of  1.62  for  the 

D  lines  and  1.63  for  the  F  line.     If  white  light  is 

incident  at  an  angle  of  45  degrees,  what  are  the  respective  angles  of 

emergence  for  these  two  colors?  Ans.  65°  19'.8;  6.6°  40'.8. 

28.  In  the  above  case,  what  is  the  angle  of  minimum  deviation  for  each 

color?     If  the  spectrometer  telescope  has  a  focal  length  of  30  cm.,  what 

is  the  length  of  the  spectrum  between  D  and  F  when  the  prism  is  set  for 

minimum  deviation  for  the  D  lines? 

Ans.  7),  48°  11'.2;  F,  49°  10'.4;  .71  cm. 

Total  Reflection.    29.  Looking  down  into  a  cylindrical  drinking  glass  partly 

filled  with  water,  one  cannot  see  external  objects 

through  the  sides  of  the  glass,  but  if  a  finger  is  firmly  pressed  against 

the  side  of  the  glass  it  can  be  seen  from  above.     Explain. 

30.  Light  incident  internally  on  the  surface  of  a  glass  prism  at  an  angle  of 

56°  is  totally  reflected  from  a  drop  of  liquid  in  contact  with  the  glass. 

If  the  index  of  refraction  of  the  latter  is  1.62  for  sodium  light,  what  is  the 

index  of  refraction  of  the  liquid?  Ans.  1.343. 

31.  In  a  system  of  Newton's  rings  due  to  a  convex  lens 
Interference.  .       ,,     „_,,  .    ,          » 

resting  on  a  plane  surface  the  25th  ring  is  1  cm.  from 

the  center,  when  sodium  light  is  used.     What  is  the  thickness  of  the  air 
film  at  that  point,  and  what  is  the  radius  of  curvature  of  the  lens? 

Ans.  0.00751  mm.;  6.67  meters. 


PROBLEMS  673 

32.  If  the  air  film  is  replaced  by  water  in  the  above  example,  what  will  be 
the  distance  of  the  25th  ring  from  the  center?  Ans.  0.97  cm. 

33.  Light  from  a  narrow  slit  passes  through  two  parallel  slits  0.2  mm.  apart. 
The  interference  bands  on  a  screen  100  cm.  away  are  2.95  mm.  apart. 
What  is  the  wave-length  of  the  light?  Ans.  0.00059  mm. 

34.  The  angles  of  a  Fresnel  biprism  are  10'  and  the  index  of  refraction  1.62. 
What  is  the  distance  between  the  two  images  of  a  slit  20  cm.  from  the 
prism?     What  is  the  width  of  the  interference  bands  of  sodium  light 
formed  on  a  screen  50  cm.  beyond  the  prism?     What  is  their  width  if 
light  of  the  wave-length  of  the  F  line  is  used? 

Ans.  0.724  mm.;  0.57  mm.;  0.47  mm. 

35.  A  film  of  glass  of  index  of  refraction  1.54  is  introduced  in  one  of  the 
interfering  beams  of  a  Michelson  interferometer,  and  causes  a  displace- 
ment of  20  fringes  of  sodium  light  across  the  field.     What  is  the  thick- 
ness of  the  film?  Ans.  0.0218  mm. 

36.  The  D  lines  in  the  spectrum  of  the  second  order  formed  by  a  Rowland 
concave  grating  of  15  feet  radius  of  curvature  are  315  cm.  from  the  slit. 
What  is  the  distance  between  rulings?  Ans.  0.00171  mm. 

«..„.      ..  37.  The  central  maximum  of  the  diffraction  bands  of 

Diffraction.  ,.       ,.  ,  ,         .       ,  , 

sodium  light  produced  by  a  narrow  slit  on  a  screen  at 

a  distance  of  100  cm.  is  2  mm.  wide.     How  wide  are  the  other  maxima 

and  the  slit?  Ans.  1  mm.;  0.589  mm. 

38    Explain  the  diffraction  bands  in  the  shadow  of  a  needle  or  wire  (Fig.  558). 

39.  Describe  and  explain  the  appearance  of  the  filament  of  a  distant  electric 
light  seen  through  very  small  pinholes  of  different  sizes. 

40.  Two  narrow  slits  0.1  mm.  apart  are  illuminated  by  sodium  light.     What 
must  be  the  diameter  of  a  lens  5  meters  away  to  clearly  resolve  the 
images  of  the  two  slits?  Ans.  2.95  cm. 

41.  In  the  above  case,  at  what  distance  will  the  same  lens  clearly  resolve 
the  images  of  the  slits  if  they  are  illuminated  by  light  of  wave-length 
corresponding  to  that  of  the  F  line?  Ans.  6.05  m. 

Polarization.      42.  Plane  polarized  light  falls  normally  on  a  plate  of 
quartz  with  faces  parallel  to  the  axis.     If  the  vibra- 
tions of  the  incident  light  are  at  an  angle  of  30°  with  the  principal  plane, 
calculate  the  relative  intensities  of  the  transmitted  ordinary  and  extra- 
ordinary rays.  Ans.  0.25,  0.75. 

43.  In  the  above  case,  if  the  crystal  is  1  mm.  thick,  what  is  the  difference  of 
phase  upon  emergence  of  the  ordinary  and  extraordinary  rays  of  sodium 
light  (§762).  Ans.  15.45  L 

44.  A  crystal  of  Iceland  spar  cut  with  faces  parallel  to  the  axis  is  2  cm.  thick. 
How  far  below  the  upper  surface  are  the  ordinary  and  extraordinary 
images  of  a  pencil  mark  on  the  lower  face?  Ans.  1.206,  1  346. 

45.  Through  how  many  degrees  will  a  column  20  cm.  long  of  a  10  per  cent, 
solution  of  cane  sugar  rotate  the  plane  of  polarization  of  sodium  light? 

Ans.  132°.88. 

Spectrum.        46.  On  mapping  the  spectral  intensity  curve  of  an  incan- 
descent source  it  is  found  that  the  maximum  intensity 

43 


674  LIGHT 

a 

is  at  a  wave-length  12,000  Angstrom  units.     What  is  the  temperature 
of  the  source  -4ns.  2^399°  abs. 

47.  The  displacement  of  the  F  line  of  hydrogen  (wave-length  4861  Angstrom 
units)  in  the  spectrum  of  a  star  is  .1  of  a  unit  toward  the  violet.  What 
are  the  direction  of  motion  and  the  velocity  of  the  star  in  the  line  of 
sight? 

k 
Ans.  6.2          toward  earth. 


Logarithms  of  Numbers  from  i  to  1000. 


No. 

o 

i 

2 

3 

4 

5 

•  6 

7 

8 

9 

10 

0000 

0043 

0086 

0128 

0170 

O2I2 

0253 

0294 

0334 

°374 

ii 

0414 

0453 

0492 

0531 

0569 

0607 

0645 

0682 

0719 

0755 

12 

0792 

0828 

0864 

0899 

0934 

0969 

1004 

1038 

1072 

1106 

13 

"39 

"73 

1206 

1239 

1271 

1303 

1335 

1367 

1399 

1430 

14 

1461 

1492 

1523 

1553 

1584 

1614 

1644 

1673 

J7°3 

1732 

15 

1761 

1790 

1818 

1847 

1875 

1903 

I931 

1959 

1987 

2014 

16 

2041 

2068 

2095 

2122 

2148 

2175 

2201 

2227 

2253 

2279 

17 

2304 

233° 

2355 

2380 

2405 

2430 

2455 

2480 

2504 

2529 

i3 

2553 

2577 

2601 

2625 

2648 

2672 

2695 

2718 

2742 

2765 

19 

2788 

2810 

2833 

2856 

2878 

2900 

2923 

2945 

2967 

2989 

20 

3010 

3032 

3054 

3075 

3096 

3Il8 

3J39 

3160 

3181 

3201 

21 

3222 

3243 

3263 

3284 

33°4 

3324 

3345 

3365 

3385 

3404 

22 

3424 

3444 

3464 

3483 

3502 

3522 

354i 

3579 

3598 

23 

3617 

3636 

3674 

3692 

37" 

3729 

3747 

3766 

3784 

24 

3802 

3820 

3838 

3856 

3874 

3892 

3909 

3927 

3945 

3962 

25 

3979 

3997 

4014 

4031 

4048 

4065 

4082 

4099 

4116 

4133 

26 

415° 

4166 

4183 

420O 

4216 

4232 

4249 

4265 

4281 

4298 

27 

43*4 

433° 

4346 

4362 

4378 

4393 

4409 

4425 

4440 

44  56 

28 

4472 

4487 

4502 

45*8 

4533 

4548 

4564 

4579 

4594 

4609 

29 

4624 

4639 

4654 

4669 

4683 

4698 

47J3 

4728 

4742 

4757 

30 

4771 

4786 

4800 

4814 

4829 

4843 

4857 

4871 

4886 

4900 

31 

4914 

4928 

4942 

4955 

4969 

4983 

4997 

5011 

5024 

5038 

32 

5051 

5°65 

5°79 

5092 

5132 

5145 

5*59 

5i72 

33 

5198 

5211 

5224 

5237 

5250 

5263 

5276 

5289 

34 

5315 

5328 

5340 

5353 

5366 

5378 

5403 

54*6 

5428 

35 

544i 

5453 

5465 

5478 

549° 

5502 

5515 

5527 

5539 

5551 

36 

5563 

5575 

5587 

5599 

5611 

5623 

5635 

5647 

5658 

5670 

37 

5682 

5694 

57°5 

5717 

5729 

5740 

5752 

5763 

5775 

5786 

38 

5798 

5809 

5821 

5832 

5843 

5855 

5866 

5877 

5888 

5899 

39 

5922 

5933 

5944 

5955 

5966 

5977 

5988 

5999 

6010 

40 

6021 

6031 

6042 

6053 

6064 

6075 

6085 

6096 

6107 

6117 

6128 

6138 

6149 

6160 

6170 

6180 

6191 

6201 

6212 

6222 

42 

6232 

6243 

6253 

6263 

6274 

6284 

6294 

6304 

6314 

6325 

43 

6335 

6345 

6355 

6365 

6375 

6385 

6395 

6405 

6415 

6425 

44 

6435 

6444 

6454 

6464 

6474 

6484 

6493 

6503 

6513 

6522 

45 

6532 

6542 

6551 

6561 

6571 

6580 

6590 

6599 

6609 

6618 

46 

6628 

6637 

6646 

6656 

6665 

6675 

6684 

6693 

6702 

6712 

47 

6721 

6730 

6739 

6749 

6758 

6767 

6776 

6785 

6794 

6803 

48 

6812 

6821 

6830 

6839 

6848 

6857 

6866 

6875 

6884 

6893 

49 

6902 

6911 

6920 

6928 

6937 

6946 

6955 

6964 

6972 

6981 

50 

6990 

6998 

7007 

7016 

7024 

7°33 

7042 

7050 

7059 

7067 

51 

7076 

7084 

7°93 

7101 

7110 

7118 

7126 

7135 

7152 

52 

7160 

7168 

7177 

7185 

7J93 

7202 

7210 

7218 

7226 

7235 

53 

7243 

72  51 

7259 

7267 

7275 

7284 

7292 

7300 

7308 

73  *6 

54 

7334 

7332 

7340 

7348 

7356 

7364 

7372 

7388 

7396 

No. 

o     i     2    3    4 

5 

6 

7 

8 

9 

675 


Logarithms  of  Numbers  from  i  to  1000. 


No. 

0 

z 

2 

3 

4 

5 

6 

7 

8 

9 

55 

7404 

7412 

7419 

7427 

7435 

7443 

7451 

7459 

7466 

7474 

56 

7482 

7490 

7497 

7505 

7513 

7520 

7528 

7536 

7543 

7551 

57 

7559 

7566 

7574 

7582 

7589 

7597 

7604 

7612 

7619 

7627 

58 

7634 

7642 

7649 

7657 

7664 

7672 

7679 

7686 

7694 

7701 

59 

7709 

7716 

7723 

7731 

7738 

7745 

7752 

7760 

7767 

7774 

60 

7782 

7789 

7796 

7803 

7810 

7818 

7825 

7832 

7839 

7846 

61 

7853 

7860 

7868 

7875 

7882 

7889 

7896 

7903 

7910 

7917 

62 

7924 

7931 

7938 

7945 

7952 

7959 

7966 

7973 

7980 

7987 

63 

7993 

8000 

8007 

8014 

8021 

8028 

8035 

8041 

8048 

8055 

64 

8062 

8069 

8075 

8082 

8089 

8096 

8102 

8109 

8116 

8122 

65 

8129 

8136 

8142 

8149 

8156 

8162 

8169 

8176 

8182 

8189 

66 

8195 

8202 

8209 

8215 

8222 

8228 

-8235 

8241 

8248 

8254 

67 

8261 

8267 

8274 

8280 

8287 

8293 

8299 

8306 

8312 

8319 

68 

8325 

8331 

8338 

8344 

8351 

8357 

8363 

8370 

8376 

8382 

69 

8388 

8395 

8401 

8407 

8414 

8420 

8426 

8432 

8439 

8445 

70 

8451 

8457 

8463 

8470 

8476 

8482 

8488 

8494 

8500 

8506 

71 

8513 

8519 

8525 

8531 

8537 

8543 

8549 

8555 

8561 

8567 

72 

8573 

8579 

8585 

8591 

8597 

8603 

8609 

8615 

8621 

8627 

73 

8633 

8639 

8645 

8651 

8657 

8663 

8669 

8675 

8681 

8686 

74 

8692 

8698 

8704 

8710 

8716 

8722 

8727 

8733 

8739 

8745 

75 

8751 

8756 

8762 

8768 

8774 

8779 

8785 

8791 

8797 

8802 

76 

8808 

8814 

8820 

8825 

8831 

8837 

8842 

8848 

8854 

8859 

3 

8865 
8921 

8871 
8927 

8876 
8932 

8882 
8938 

8887 
8943 

8893 
8949 

8899 
8954 

8904 
8960 

8910 
8965 

8915 
8971 

79 

8976 

8982 

8987 

8993 

8998 

9004 

9009 

9015 

9020 

9025 

80 

9031 

9036 

9042 

9047 

9053 

9058 

9063 

9069 

9074 

9079 

81 

9085 

9090 

9096 

9101 

9106 

9112 

9117 

9122 

9128 

9*33 

82 

9138 

9M3 

9149 

9154 

9159 

9165 

9170 

9r75 

9180 

9186 

83 

9191 

9196 

9201 

9206 

9212 

9217 

9222 

9227 

9232 

9238 

84 

9243 

9248 

9253 

9258 

9263 

9269 

9274 

9279 

9284 

9289 

85 

9294 

9299 

9304 

9309 

9315 

9320 

9325 

9330 

9335 

9340 

86 

9345 

9350 

9355 

9360 

9365 

937° 

9375 

938o 

9385 

9390 

87 

9395 

9400 

9405 

9410 

941  5 

9420 

9425 

9430 

9435 

9440 

88 

9445 

945° 

9455 

9460 

9465 

9469 

9474 

9479 

9484 

9489 

89 

9494 

9499 

9504 

9509 

9518 

9523 

9528 

9533 

9538 

90 

9542 

9547 

9552 

9557 

9562 

9566 

9571 

9576 

958i 

9586 

91 

9590 

9595 

9600 

9605 

9609 

9614 

9619 

9624 

9628 

9633 

92 

9638 

9643 

9647 

9652 

9657 

9661 

9666 

9671 

9675 

9680 

93 

9685 

9689 

9694 

9699 

9703 

9708 

9713 

9717 

9722 

9727 

94 

9731 

9736 

974i 

9745 

975° 

9754 

9759 

9763 

9768 

9773 

95 

9777 

9782 

9786 

9791 

9795 

9800 

9805 

9809 

9814 

9818 

96 

9823 

9827 

9832 

9836 

9841 

9845 

9850 

9854 

9859 

9863 

97 

9868 

9872 

9877 

9881 

9886 

9890 

9894 

9899 

9903 

9908 

98 

9912 

9917 

9921 

9926 

993° 

9934 

9939 

9943 

9948 

9952 

99 

9956 

9961 

9965 

9969 

9974 

9978 

9983 

9987 

9991 

9996 

No. 

o 

I 

2 

3 

4 

5 

6 

7 

8 

9 

676 


Natural  Sines  and  Cosines. 


Sine 


Cosine 


4  o 

0.0000 

90 

1.  0000 

i 

0.0175 

175 

89 

0.9998 

02 

2 

0.0349 

174 

88 

0.9994 

04 

3 

0.0523 

174 

87 

0.9986 

08 

4 

0.0698 

175 

86 

0.9976 

10 

5 

0.0872 

174 

85 

0.9962 

H 

6 

0.1045 

173 

84 

0.9945 

I7 

7 

0.1219 

174 

83 

0.9925 

so 

8 

0.1392 

J73 

82 

0.9903 

22 

9 

o.i  564 

172 

81 

0.9877 

26 

10 

0.1736 

172 

80 

0.9848 

29 

ii 

0.1908 

172 

79 

0.9816 

32 

12 

0.2079 

171 

78 

0.9781 

35 

J3 

0.2250 

171 

77 

0.9744 

37 

14 

0.2419 

169 

76 

0.9703 

4i 

15 

0.2588 

169 

75 

0.9659 

44 

16 

0.2756 

168 

74 

0.9613 

46 

*7 

0.2924 

1  68 

73 

0.9563 

50 

18 

0.3090 

166 

72 

0.9511 

52 

19 

0.3256 

166 

71 

0-9455 

56 

20 

0.3420 

164 

70 

0-9397 

58 

21 

0.3584 

164 

69 

0.9336 

61  • 

22   - 

0.3746 

162 

68 

0.9272 

64 

23 

0.3907 

161 

67 

0.9205 

•   67 

24 

0.4067 

1  60 

66 

0-9135 

70 

25 

0.4226 

J59 

65 

0.9063 

72 

26 

0.4384 

158 

64 

0.8988 

75 

27 

0.4540 

156 

63 

0.8910 

78 

28 

0.4695 

155 

62 

0.8829 

81 

29 

0.4848 

153 

61 

0.8746 

83 

3° 

0.5000 

152 

60 

0.8660 

86 

3i 

0.5150 

I5° 

59 

0.8572 

88 

32 

0.5299 

149 

58 

0.8480 

92 

33 

0.5446 

J47 

57 

0.8387 

93 

34 

0.5592 

146 

56 

0.8290 

97 

35 

0-5736 

144 

55 

0.8192 

98 

36 

0.5878 

142 

54 

0.8090 

IO2 

37 

0.6018 

140 

53 

0.7986 

104 

38 

0.6157 

139 

52 

0.7880 

106 

39 

0.6293 

136 

5i 

0.7771 

109 

40 

0.6428 

135 

5° 

0.7660 

in 

4i 

9.6561 

133 

49 

0.7547 

"3 

42 

0.6691 

130 

48 

0.7431 

116 

43 

0.6820 

129 

47 

0.7314 

117 

44 

0.6947 

127 

46 

0.7*93 

121 

45 

0.7071 

124 

45°  * 

0.7071 

122 

Cosine 


Sine 


D  10 


677 


INDEX  TO  NAMES 


Numbers  Refer  to  Pages. 


Amagat,  152.  221 

Ampere,  434,  465,  466 

Andrews,  254,  258,  259 

AngstrSm,  639 

Arago,  444,  470 

Archimedes,  126,  133,  134,  148 

Argand,  631 

Arons,  475 

Avogadro,  225 

Barlow,  467,  468 

Barnes,  232 

Becquerel,  502,  503,  505,  508 

Beethoven,  518 

Bell,  462,  463 

Bernoulli,  135,  136 

Beasel,  30,  101 

Bigelow,  333 

Billet,  604  j 

Biot,  381 

Blondel,  461 

Boltzmann,  274 

Bourdon,  154 

Boyle,  151,  152,  153,  154,  155,  157,  223, 

226,  237,  283,  520,  605 
Boys,  103,  120,  423 
Bradley,  563,  566 
Bragg,  495 
Branly,  477 
Braun,  355,  462,  472 
Brewster,  655 
Bridgman,  242 
Broca,  388 
Brodhun,  631 
Bunsen,  150,  159,  232,  502,  541,  554, 

561,  602,  636,  637,  638,  643 

Callendar,  211,  279,  395 

Carnot,  286,  287,  288,  289,  290,  293, 

297,  307 
Cauchy,  667 

Cavendish,  102,  103,  120,  348 
Charles,  222,  223 
Clark,  415 
Claude,  261 
Olausius,  288,  289,  293 


224, 


560, 


295, 


Cornu,  564.  566 

Coulomb,  90,  318,  319,  320,  321,  323,  324, 

348,  365 

Crookes,  462,  490,  508 
Curie,  Mme.,  503,  504 
Curie,  M.,  504,  513 


Danieil,  412,  413.  414 

D'Arsonval,  384,  388,  389,  390,  423,  444 

Davy,  200 

Debierne,  504 

De  la  Tour,  254 

Descartes,  566 

Dewar,  261 

Diesel,  302 

Dolezalek,  355 

Dollond,  588 

Doppler,  530,  643,  644 

Draper,  634 

Duddell,  423,  461,  471 

Dulong,  219,  239 


Einthoven,  389 

Ewing,  312,  427,  436,  471 


Fahrenheit,  203.  204 

Faraday,  258,  312,  314,  316,  343,  344,  345, 
347,  364,  382,  409,  416,  431,  433,  437,  438, 
440,  441,  443,  444,  445,  448.  467,  468,  490, 
666 

Faure,  416 

Feddersen,  472 

Ferraris,  469 

Fdry,  276 

Fizeau,  564,  565,  566,  582 

Fleming,  441,  451.  464 

Foley,  522 

Forbes,  444,  566 

Fortin,  149,  150 

Foucault,  166,  444,  564,  565,  566,  582,  658 

Fourier,  186 

Franklin.  338,  363,  370 

Fraunhofer,  587,  588,  589,  641,  642.  647,  663 

Fresnel,  604,  605,  651,  655,  663,  664 


679 


680 


INDEX  TO  NAMES 


Gaede,  162,  163 

Galileo,  21,  26,  30,  101,  136,  149,  207,  562. 

618,  624 
Galvani,  412 
Gauss,  323,  324 
Gay-Lussac,  222,  235 
Gilbert,  329,  336,  337 
Gramme,  457 
Gray,  337 
Guericke,  161 

Haidinger,  608 
Hampson,  260 
Hartmann,  330 
Hefner,  631 

HelmholtB,  280,  386,  411,  446,  535 
Henry,  437,  471 
Hersohel.  625,  632 
Hertf ,  470,  473,  474,  475,  476,  481 
Heusler,  435 
Holborn,  277 
Holtz,  357 

Hooke,  112,  113,  118,  122 
Hope,  220,  221 
Hop  kins  on,  433 
Hughes,  462 
Hunning,  463 

Huyghens,  568,  571,  582,  622,  650,  652,  656, 
659 

Joly,  232,  561 

Joule,  201,  235,  238,  277,  280,  402,  405 

Kater,  86 

Kaufmann,  505 

Kelvin,  171,  238,  239,  288.  289,  290,  354, 

370,  386,  422,  465,  471,  472 
Kepler,  100 
Kerr,  661.  666 

Kirchhoff,  273,  398,  636,  641,  642,  653 
Kleomedes,  559 
Koenig,  647 
Kohlrausoh,  408,  482 
Kundt,  545,  551 

Laborde,  513 

Lambert,  561 

Landolt,  665 

Langley,  643 

Langmiur,  162 

Laplace,  520 

Le  Chatelier,  424 

Lecher,  475 

Leclanohe,  412,  414 

Lena,  389,  435,  440,  444,  470 

Linda   260,  261 

Lippmann,  609 


Lissajoua,  181 
Lloyd,  604 
Lodge,  471,  473 
Lorentz,  666 
Lummer,  631 
Lyman,  639 

Malus,  653 

Marconi,  471,  476,  477 
Mariotte,  151 
Marx,  494 

Maxwell,  312,  314,  349,  369,  375,  382,  430, 
431,  434,  448,  463,  466,  471,  474,  481,  482 
Mayer,  237 
Methven,  631 

Michelson,  564,  565,  566,  629,  630 
Miller,  522,  533,  535 
Milliken,  501 
Morae,  277 

Nernst,  239,  403,  417 

Newcomb,  564,  565,  566 

Newton,  2,  5,  26,  28,  29,  31,  32,  34,  99,  100. 
101, 103, 116,  274,  343,  520,  567,  568,  587, 
605,  606,  607,  608,  619,  625,  646,  657,  662 

Nicholson,  134 

Nicol,  658,  659,  662,  663 

Oersted,  374,  382 

Ohm,  392,  393,  398,  399,  402,  403,  407,  408. 

418,  432,  442,  497,  498 
Olzewski,  261 
Omnes,  262 

Pascal,  125,  126,  148,  149 

Peltier,  422 

Perrin,  491 

Petit,  219,  239 

Pictet,  260 

Phot,  137 

Planck,  636 

Plante,  416 

Poulsen,  479 

Prevost.  271 

Ramsden,  622 

Rankine,  296 

Rayleigh,  523 

Regnault.  223,  248 

RSntgen,  493,  494,  495,  496,  497.  498,  502, 

503 

Roget,  467 
ROmer,  562,  563 

Rowland,  228,  277,  279,  383,  384,  629,  642 
Rumford,  200,  560 
Rutherford,  504,  506,  507,  509.  510,  512 


INDEX   TO  NAMES 


681 


Sabine,  546,  547,  548,  549.  550 

Savart,  381,  528 

Schmidt,  503 

Schumann,  639 

Seebeck,  419 

Seibt,  475 

Siemens,  457,  466 

Snell,  582,  583 

Soddy,  509 

Souder,  522 

Stefan,  274,  306.  636 

Stevinus,  94 

Stokes,  501,  641,  648 

Symner,  338 

Tammann,  222,  242 

Tesla,  450,  469,  471,  475 

Thompson,  358 

Thomson,  E.,  479 

Thomson  (see  Kelvin) 

Thomson,  J.  J.,  268.  369,  382,  491,  492,  493, 

501,  513 

Thomson,  J.,  241,  257 
Thwing,  276 
Toepler,  357 
Torricelli,  136,  149 
Tripler,  260 


Tyndall,  422,  860 

Van  der  Waals,  153,  157,  257,  258 

Venturi,  128 

Verdet,  666 

Violle,  404 

Volta,  356,  372,  373,  412,  416,  417 

Wanner,  277 

Wagner,  449 

Watt,  34 

Weber,  482 

Wehnelt,  449 

Wehrsen,  359 

Weston,  390,  415 

Welsbach,  521 

Wheatstone,  210,  272,  400,  401,  448 

Wien,  636 

Wiener,  609,  654 

Wimshurst,  357,  359 

W6llaston,  587 

Young,   114,  115.  188,  566,  568.  569,   804, 
606,  607,  608,  616,  647 

Zeeman,  666,  669 
Zenneck,  472 


INDEX  TO  SUBJECTS 


Numbers  Refer  to  Pages. 


Aberration,  astronomical,  563 

chromatic,  598 

spherical,  580,  595-598 
Absolute  temperature,  206 

units,  104-106,  480-482 

zero,  204 

Absorption  of  a-rays,  507 
radiant  energy,  272,  640 
Absorbing  power,  272 
Acceleration,  angular,  52 

linear,  18-26,  59 
Achromatic  prism,  590 

combinations,  598 
Actinium,  504 
Action  and  reaction,  32 
Activity,  44 

excited,  511 
Adiabatic  processes,  283 

elasticity,  285,  520 

equation  of,  283 
Air,  manometer,  154 

pump,  mechanical,  161 

pump,  mercury,  162,  163 
Alloys,  resistance  of,  394 
Alloys  and  solutions,  217,  244,  267 
Alpha  rays,  506 
Alternating  currents,  452 
Ammeter,  390,  399 
Ampere,  the,  378,  480 
Ampere's  electrodynamic  apparatus,  466 

theory  of  magnetism,  434 
Amplitude  of  S.H.M.,  79 
Analytical  method,  37 
Aneroid  barometer,  150 
Angle  of  capillarity,  142 

critical,  600 

of  minimum  deviation,  585 

polarizing,  655 
Angular  acceleration,  52 

harmonic  motion,  83 

momentum,  65 

velocity,  51 
Anion,  406 
Anode,  405 

Apparent  expansion  of  liquids,  219 
Arago's  experiment,  444 


Arc  lamp,  404 

Archimedes,  principle.  126,  148 

Armatures,  457 

Ascent  of  liquids  in  capillary  tubes,  142 

Aspirator,  128 

Astatic  needle,  387 

Astigmatic  pencil,  582 

Astronomical  telescope,  624 

Atmosphere,  pressure  of,  148 

Atmospheric  electricity,  370 

Atoms,  107,  201 

Attraction  and  repulsion,  electrical,  335 

magnetic,  310 
Audibility,  limits  of,  529 
Aurora  borealis,  371 
Avogadro's  law,  225 
Axes  of  lens,  594 

of  crystal,  649 


Balance,  30 

torsion,  319,  348 

wheel  of  watch,  217 
Ballistic  galvanometer,  390 
Barlow's  wheel,  467 
Barometers,  aneroid,  150 

mercury,  149 
Baseball,  curve  of,  129 
Beats,  179,  185,  531,  535 
Bells,  544 

Bernoulli's  theorem,  135 
Beta  rays,  505 
Biaxial  crystals,  652 
Biot  and  Savart's  law,  381 
Black  body,  273 
Block  and  tackle,  97 
Boiling-point,  248 

influence  of  dissolved  substances  on,  250 
Bolometer,  271 

Bourdon's  pressure  gauge,  154 
Boyle's  law,  151,  155,  157,  224 
Braun  tube,  462 
Brewster's  Law,  655 
Bridge,  Wheatstone's  210,  400 
British  thermal  unit,  227 
Buoyancy,  126,  153 


683 


684 


INDEX  TO  SUBJECTS 


Caloric  theory  of  heat,  199 

Calorie,  227 

Calorimeter,  229 

Calorimetry,  226 

Camera,  photographic,  625 

Capacity,  electrostatic,  359-36*6 

thermal,  228 

unit  of,  359,  365 
Capillarity,  142 
Carnot's  cycle,  286-288 
Cathode  rays,  490 
Caustic,  by  reflection,  580 

by  refraction,  595 
Cavendish  experiment,  102 
Cells,  voltaic,  371,  373,  411-419 

secondary,  415 

standard,  415 

Centigrade  thermometer  scale,  203 
Center  of  buoyancy,  126 

gravity,  72 

mass,  57-61 

percussion,  86 

oscillation,  86 

optical,  594 
Centrifugal  couple,  73 

cream  separator,  34 

drier,  34 

force,  33,  72 
Chain  hoist,  97 
Change  of  state,  240-262 
Charge,  electrical,  energy  of,  347 

residual,  362 
Charles'  law,  222 
Chromatic  aberration,  598 
Clark  cell,  415 

Claude's  method  of  liquefaction,  261 
Clock  pendulum,  217 
Coefficient  of  expansion,  cubical,  218 

linear,  215 
Coil,  induction,  448 
Cold  produced  by  evaporation,  251 

expansion  of  gases,  260 
Colloids,  146 
Color,  570,  644-648 

blindness,  647 
Color  sensation,  646 
Colors,  complementary,  6*6 

primary,  647 

pigment,  648 
Combustion,  heat  of,  239 
Commutator,  455 
Complementary  colors,  645 
Composition  of  forces,  35 
Compound  microscope,  622 

pendulum,  85 

wound  dynamo,  457 
Compounds,  107 


Compressibility  of  liquids,  132 
Concave  mirror,  575 

lens,  590 
Condenser,  electrical,  361-368 

oscillatory  discharge  of,  471 
Conductance,  392 
Conduction,  electrolytic,  405 

in  gases,  489 

thermal,  262,  264 
Conductivity,  electrical,  393 

thermal,  264 
Conductors,  336 
Conjugate  foci,  573,  591 
Conservation  of  electricity,  347 

energy,  50,  277 
Constant,  dielectric,  364 
Constitution  of  matter,  106-108 
Contact  difference  of  potential,  372,  417 
Continuity  of  state,  257 
Convection  of  heat,  262 
Convex  lens,  590 

mirror,  577 

Cooling,  Newton's  law  of,  274 
Cords,  vocal,  535 
Cornea,  618 

Corpuscular  theory  of  light,  567 
Corresponding  states,  258 
Coulomb's  balance,  319,  348 
Coulomb's  law,  electro-static,  348 

magnetic,  318 
Coulomb,  the,  379,  480 
Coulometer,  411 
Couples,  71,  73 

Couple,  thermo-,  271,  419-424,  633 
Critical  angle,  600 

pressure,  256 

temperature,  254,  256 
.  volume,  256 
Cryohydrate,  244 
Crystalloids,  146  . 

Crystals,  biaxial,  652 

positive  and  negative,  652 

uniaxial,  652 
Current,  the  electric,  371 
Currents,  action  of  magnets,  on,  464 

alternating,  452,  444 

Foucault,  or  eddy,  444 

primary  and  secondary,  439 
Curvilinear  motion,  13 
Cyclic  operations,  285 
Cylindrical  lens,  598 

mirror,  581 


Damping,  389 

Daniell  cell,  413 

D'Arsonval  galvanometer,  388 


INDEX  TO  SUBJECTS 


685 


Davy's  experiment  on  the  nature  of  heat, 

200 

Declination,  magnetic,  330 
Degrees  of  freedom  of  a  body,  7 
Degree,  temperature,  203 
Density,  108 

of  earth,  103 

of  water,  221 

Depression  of  freezing  point,  242 
Derived  units,  dimensions  of,  104 
Deviation,  585 
Dewpoint,  248 
Diagram  of  work,  42 
Diamagnetic  substances,  318,  433 
Dichromatism,  645 
Dielectric,  344 

constants,  364 
Diesel  motor,  302 
Diffraction,  195,  609 

grating,  616 
Diffusion  of  gases,  158 

ions,  496 

liquids,  145 

metals,  108 
Diffusivity,  146 
Dimensional  formula?,  105 
Dimensions     of    electrical    and     magnetic 
units,  480-482 

of  units,  104 

mechanical  units,  104-105 
Dip,  330 

Direct  vision  prism,  590 
Direction  of  sound,  526 
Discharge  of  condenser,  471 

in  gases,  489 

Disintegration  theory  of  radioactivity,  513 
Disk  dynamo,  443 
Dispersion,  586,  589,  667-669 
Dispersive  power,  587 
Displacement,  8-10,  79 

angular,  51 

of  spectral  line,  643 
Dissociation,  electrolytic,  244,  407 
Doppler's  effect,  530,  643 
Double  refraction,  649-666 

produced  by  strain,  660 
Drum  armature,  457 
Ductility,  120 
Dutch  telescope,  624 
Dynamics,  6,  26 

Dynamo-electric  machines,  455-460 
Dyne,  31 

Earth,  density  of,  103 

inductor,  452 

Earth's  magnetism,  329-334 
Echo,  547 


Eddy  currents,  444 
Efficiency,  94 

of  Carnot's  cycle,  288 

steam  engine,  298 
Einthoven  galvanometer,  389 
Efflux  of  gases,  158 
Elastic  fatigue,  119 

limit,  118 

waves,  187 
Elasticity,  46,  110 

modulus  of,  113 

of  gases,  153,  285 

liquids,  132 

volume,  of  solids,  113 
Electric  current,  371 

unit,  378 

field.  343 

lines  of  force,  343 

oscillations,  470-480 

quantity  unit,  379 

wave  telegraphy,  477 
Electrical  machines,  357 

potential,  351-353 

units,  480-482 

waves,  470-480 
Electricity,  theories  of,  338 
Electrification  by  induction,  337 
Electro-chemical  equivalent,  409 

dynamics,  463-470 

dynamometer,  466 
Electro-magnetic  induction,  437 
theory  of  light,  474 
units,  378,  480-482 
waves  along  wires,  475 
waves,    reflection    and    refraction 

of,  473 

waves,  stationary,  475 
Electromotive  force,  372,  390 

units  of,  390 
Electrons,  107,  201,  268,  339,  347,  383,  402, 

405,  411,  434,  493,  505 
Electro-static  induction,  337 
machines,  357 
units,  348,  481-484 
Electrodes,  405 

polarization  of,  407 
Electrolysis,  405, 
Electrolyte,  373,  405,  409 

dissociation  of,  407 
Electro  magnet,  424 
Electrometers,  354-355 
Electromotive  force,  372,  390 
Electrophorus,  356 
Electroscope,  gold  leaf,  339 
Elements,  107 

Emanations  from  radio-active  bodies,  510 
Emission  of  radiant  energy,  272 


686 


INDEX  TO  SUBJECTS 


Emission  theory  of  light,  567 

Emiasivity,  272,  641 

Energy,  40-51,  77,  80,  86,  118 

conservation  of,  51,  279 

kinetic,  44,  201 

of  charge,  347,  366 

of  rotation,  63 

potential,  47,  201 

radiant,  269 
Engine,  internal  combustion,  300 

steam,  295-299 

turbine,  299 
Entropy,  292-294 
Equation  of  state,  213 
Equilibrium  of  a  body,  73-78 

of  a  particle,  39 

stable  and  unstable,  76 
Equipotential  surfaces,  351 
Equivalent  air-path,  591 

electro-chemical,  409 
Erg.  42 

Ether,  the,  571 
Eutectic  alloy,  244 
Exchanges,  theory  of,  270 
Excited  radioactivity,  511 
Expansion,  absolute,  of  mercury,  219 

liquids,  218 

of  gases,  222,  238 

of  solids,  214-218 

real  and  apparent,  219 
Swing's  model  of  magnet,  312 
Eye,  617 

pieces,  622 

Fahrenheit's  thermometric  scale,  203 
Farad,  the,  365 
Faraday  disk  dynamo,  443 
Faraday's  induction  experiments,  437 
Faraday's  ice-pail  experiment,  345 
Faraday's  laws  of  electrolysis,  409 
Fatigue,  elastic,  119 
Ferromagnetic  substances,  318,  433 
Field,  electric,  343,  368,  369 

magnetic,  312,  315,  320,  436 

magnets,  456 
Fizeau's   measurement   of  the   velocity    of 

light,  564 
Flexure,  115 
Flotation,  134 
Fluids,  106,  120-132 
Fluorescence,  633,  648 
Focal  lines,  581,  597 
Foci,  conjugate,  573,  591 
Focus,  194 
Foot-pound,  42 
Force,  26-40,  79 

centrifugal,  33,  72 


Force,  parallelogram,  35 

unit  of,  30 
Forces,  composition  of,  35 

conservative,  60 

molecular,  107 

moment  of,  62 

parallel,  69 

parallelogram  of,  35 

polygon  of,  36 

resolution  of,  36 

resultant  of,  35,  68 

triangle  of,  35 
Fortin's  barometer,  150 
Foucault's  currents,  444 

measurement     of     velocity     of     light, 
564 

prism,  658 

Fourier's  theorem,  186 
Fraunhofer  lines,  587,  641 
Freezing  point,  241 

depression  of,  244 
influence  of  pressure  on,  241 
Frequency  of  notes,  528 

of  waves,  182 

Fresnel's  experiments  on  the  interference  of 
light,  604 

rhomb,  663 
Friability,  120 
Friction,  static,  88 

kinetic,  91 

rolling,  92 
Fusion,  241 

heat  of,  244 

Gaede  air-pump,  162 
Galiliean  telescope,  624 
Galvanometer,  D'Arsonval,  388 

ballistic,  390 

Broca,  387 

Einthoven,  389 

Thomson,  386 

tangent,  384 
Gamma  rays,  508 
Gas  law,  223 

perfect,  223,  291 

thermometer,  202 
Gases,  148-163 

diffusion  of,  158 

efflux  of,  158 

expansion  of,  222,  238 

kinetic  theory  of,  155,  233 

liquefaction  of,  258 

perfect,  223,  291 

specific  heat  of,  233 

thermal  conductivity  of,  267 

velocity  of  sound  in,  519,  545 

viscosity  of,  154 


INDEX  TO  SUBJECTS 


687 


Gause,  the,  320 
Goldleaf  electroscope,  339 
Gramme  armature,  457 
Graph  of  a  velocity,  14 

work  done,  42 

Grating  diffraction,  616,  629 
Gravitation,  99,  104 
Gravitational  waves,  192 
Gravity,  45 

center  of,  72 

value  of  at  different  latitudes,  22 
Gyration,  radius  of,  64 
Gyroscope,  87 


Hardness,  120 

Harmonics,  533 

Heat,  absorption  of,  272 

capacity,  228 

conduction  of,  264 

convection  of,  262 

emission  of,  272 

engine,  295 

mechanical  equivalent  of,  227,  277 

of  combustion,  239 

of  fusion,  244 

of  vaporization,  250 

radiation  of,  269 
Heat,  specific,  227 

units  of,  226 

Heating  by  electric  currents,  402-405 
Height  by  barometer,  151 
Helium,  liquefaction  of,  262 
Helmholtz's  equation,  446 
Henry,  the,  447,  481 
Hertz's  experiments,  473 
Heusler's  alloys,  435 
Homocentric  pencil,  582 
Homogeneity,  109 
Hooke's  law,  112 
Hope's  experiment,  220 
Horse-power,  44 
Humidity  of  air,  247 
Huyghens*  principle,  571 

wave-surface  in  uniaxial  crystals,  652 
Hydraulic  press,  126 

ram,  138 

Hydrogen  thermometer,  203 
Hydrometers,  133 
Hygrometry,  248 
Hysteresis,  435 


Ice,  lowering  of  melting-point  of,   by  pres- 
sure, 241 
Iceland  spar,  649 
Ice-pail  experiment,  345 


Images  by  lens,  594 

by  mirrors,  574,  579 
Impact,  oblique,  117 

of  elastic  bodies,  110 
Impedance,  461 
Impulse,  32 

Incandescent  electric  lamp,  403 
Inclination,  330 
Inclined  plane,  98 
Index  of  refraction,  582,     588 

measurement  of,  585,  586 
Indicator  card,  296 
Induced  charge,  443 

current,  437,  443 

electromotive  force,  442,  450 
Induction,  coil,  448 

electromagnetic,  437 

coil,  Tesla,  450 

electrostatic,  337 

magnetic,  317 

Inductive  capacity,  specific,  364 
Inductor,  earth,  452 
Inertia,  27 

moment  of,  64-67 
Insulators,  336 
Intensity  of  light,  559 

magnetic  field,  320,  331 

magnetization,  425 

wave  motion,  197 
Interference  of  waves,  184 

light,  569,  604-609,  660 

sound,  535 

Interferometer,  Michelson's,  629 
Intermediate  magnetic  poles,  318 
Internal  energy,  202,  251,  281 
Intervals,  musical,  531 
Interrupter,  Wehnelt,  449 
Invar,  217 
lonization  by  or,  0,  and  y  rays,  502-506 

collision,  499 

flames,  502 

hot  metals,  501 

Rdntgen  rays,  496 

ultra-violet  light,  502 

theory  of  gases,  498 
Ions,  406,  498 

charge  on,  410,  500 

diffusion  of,  499 
Irradiation,  620 
Irrational  dispersion,  589 
Irreversible  cycle,  289 
Tsoclinic  lines,  331 
Isodynamic  lines,  333 
Isogonic  lines,  331 
Isothermal  curves,  225,  254 

process,  281 
Isotropy,  109,  216 


688 


INDEX  TO  SUBJECTS 


Jar,  Ley  den,  361 
Joule,  the,  42,  301 

Joule's  determination  of  mechanical  equiva- 
lent, 277 
experiments  on  the  expansion  of  gases, 

235,  238 
law,  402 

Kater's  pendulum,  86 

Kathode,  405 

Kation,  406 

Kelvin's  absolute  scale  of  temperature,  290 

Kepler's  laws,  100 

Kerr's  experiments  on  double-refraction  in 

dielectrics,  666 
Kinematics,  5 
Kinetics,  6 

Kinetic  energy,  44,  63,  66 
Kinetic  theory  of  gases,  155,  201 

matter,  108,  201,  207 
KirchhoflE's  law  of  radiation,  273,  641 

laws  of  electric  circuits,  398 
Kite,  37 
Kundt's  method,  545 

Lambert's  law,  561 
Lamp,  arc,  403 

incandescent,  403 
Laplace's  calculation  of  velocity  of  sound  in 

gases,  520 

Lantern,  projection,  627 
Law,  physical,  5 
Leclanchg  cell,  414 
Length,  units  of,  8 
Lens,  590-599 

achromatic,  598 

axes  of,  594 

cylindrical,  598 
Lena's  law,  440 
Level  of  surface  of  fluid,  124 
Lever,  94 
Leyden  jar,  361 

separable,  363 
Light,  diffraction,  609 

dispersion  of,  586,  589,  667 

double  refraction  of,  649-666 

emission  theory,  567 

intensity  of,  559 

interference  of,  569,  604-609,  660 

Maxwell's  theory  of,  474 

polarized,  649-666 

reflection  of,  557,  572-582* 

refraction  of,  559,  582-590 

sources  of,  554 

total  reflection,  599 

velocity  of,  562-566 

wave  theory,  568 


Light,  wave  length,  570,  616 
Lightning  rods,  370  » 

Limits  of  audibility,  529 

elasticity,  118 

Linde's  method  for  liquefaction  of  gases,  260 
Linear  expansion,  214 
Lippmann's  color-photography,  609 
Liquefaction  of  gases,  258 
Liquids,  132-148 

compressibility  of,  132 

elasticity  of,  132 
Liquids,  velocity  of  outflow  of,  136 

sound  in,  545 
Lissajous'  figures,  181,  185 
Lloyd's  mirror,  604 
Lodestone,  309 
Loop.  191 
Loudness,  527 
Luminescence,  641 
Luminosity,  standards  of,  631 
Lummer-Brodhun  photometer,  631 

Machines,  simple,  93-99 

dynamo-electrical,  455-460 

electrical,  357 
Magnetic  attraction  and  repulsion,  310 

maps,  331 

effect  of  current,  374-382 

field,  312,  315,  320,  436 

measurement  of,  323-329 
intensity  of,  320,  331 

flux,  430 

induction,  428 

lines  of  force,  312,  320 

moment,  322 

pole,  309 

pole  unit,  319 

rotation  of  plane  of  polarization,  666 

shielding,  432 

substances,  317 
Magnetizing  force,  431 
Magnetism,  309 

induced,  317 

terrestrial,  329-334 
Magnetization,  method  of,  316 

curves,  426-435,  453,  453 
Magneto-electric  machine,  456 
Magnetometer,  327 
Magnets,  309 
Magnification,  580 
Malleability,  120 
Manometer,  154 
Mass,  26-34 

center  of,  57-61 

units  of,  28 
Matter,  1,  106 

states  of,  106 


INDEX  TO  SUBJECTS 


689 


Maximum  thermometer,  209 
Maxwell's  electro-magnetio  theory  of  light, 
474 

rule,  463 

Measurements,  optical,  617 
Mechanical  equivalent  of  heat,  227,  277 
Melting-point,  241 

depression    of,    produced    by    dis- 
solved substances,  244 
influence  of  pressure  on,  241 
Mercury  pump,  162 

thermometer,  207 
Metacenter,  135 
Meter,  8 
Method  of  continuous  flow,  232 

of  mixtures,  229 
Michelson's  interferometer,  629 

measurement  of  the  velocity  of  light, 

564 
Microscope,  compound,  622 

simple,  620 

Minimum,  deviation,  585,  603 
Minimum  thermometer,  209 
Mirage,  601 
Mirrors,  concave,  575 

convex,  577 

cylindrical,  581 

hyperboloidal,  582 

paraboloidal,  582 

plane,  572 

rotating,  557 

spherical,  aberration  in,  580 
Mixtures,  method  of,  heat,  229 
Modulus  of  elasticity,  113,  153 
Molecules,  107,  201 
Molecular  forces,  139,  239 

theory  of  heat,  201 

of  magnetism,  311 
Moment  of  a  force,  62-68 

inertia,  62-68,  85 

magnet,  322 
Momentum,  29 

angular,  65 
Moon,  motion  of,  100 
Motion,  curvilinear,  13 

in  a  circle,  23,  80 

Newton's  laws  of,  26 

simple  harmonic,  78,  177 

uniformly  accelerated,  19 
Motors,  electric,  468 
Multiple  image  formed  by  mirrors,  574 
Musical  intervals,  531 

scale,  531 

Needle,  astatic,  387 

dipping,  330 
Nernst  lamp,  403 


Neutral  point,  419 

Newcomb's  measurement  of  the  velocity  of 

light,  564 
Newton's  emission  theory,  567 

law  of  cooling,  274 

law  of  gravitation,  99 

laws  of  motion,  26-33 

rings,  605 

telescope,  625 

Nicholson's  hydrometer,  133 
Nicol's  prism,  658 
Nodes  and  loops,  191 
Notes,  musical,  frequency  of,  528 

Occlusion  of  gases,  157 
Octave,  531 

Ohm,  the,  392,  396,  480 
Ohm's  law,  391,  407 
Oil  immersion,  597 
Opacity,  558 
Optical  center,  594 

instruments,  617 
Organ  pipes,  541-543 
Oscillograph,  461 
Oscillation,  center  of,  86 
Oscillatory  discharge,  470 
Osmosis,  147 
Overtones,  533,  537,  542 

Paraboloidal  mirror,  582 
Parallel  forces,  69 
Parallax,  555 
Parallelogram  method,  10 

of  forces,  35 

Paramagnetic  bodies,  318,  433 
Pascal's  principle,  125,  148 
Peltier  effect,  422 
Pendulum,  compound,  85 

double,  181 

Kater's,  86 

simple,  82 

torsion,  84 
Penumbra,  555 
Percussion,  86 
Period,  78 

of  S.  H.  M.,  81 
Periodic  motion,  78-88 
Permeability,  magnetic,  430 
Phase,  of  S.  H.  M  ,  81,  178 
Phonodeik,  522 
Phonograph,  526 
Phosphorescence,  649 
Photoelectric  effect,  501 
Photographic  camera,  625 
Photography,  color,  609 
Photography  of  sound  waves,  522,  546 
Photometry,  560,  631 
Piezometer,  133 


690 


INDEX  TO  SUBJECTS 


Pigment  colors,  648 
Pinhole  images,  556 
Pipes,  organ,  541-543 
Pitch,  528 
Pitot  tube,  137 
Plasticity,  120 
Plates,  thick,  608 

thin,  colors  of,  605 

vibrations  of,  544 
Platinum  thermometer,  210 
Polarization  of  electrodes,  407 
Polarized  light,  649-666 

rotation  of  plane  of,  663-666 

waves,  183,  190 
Polariscope,  658 
Polarizing  angle,  655 
Poles,  intermediate,  318 

magnetic,  of  the  earth,  329 

of  a  magnet,  309 
Polonium,  504 

Porous-plug  experiment,  238,  292 
Position,  7 
Positive  crystal,  652 
Potential,  electrical,  349-353 

energy,  47,  78 

of  electric  charge,  347,  366 
Potentiometer,  399 
Poundal,  31 
Power,  44 

Practical  system  of  electrical  units,  480-482 
Precession,  87 
Pressure,  effect  on  fusion,  241 

exerted  by  a  fluid,  121,  137 

gauge,  154 

of  gases,  212 
Pressure  of  the  atmosphere,  148 

within  a  soap-bubble,  144 
Prevost's  law  of  exchanges,  271 
Primary  colors,  647 
Prism,  585,  590 

Foucault,  658 

minimum  deviation  of,  585 

Nicol's,  658 

uniaxial,  652 
Projectile,  22 
Projection  lantern,  627 
Pulley,  96 
Pumps  for  gases,  161-163 

liquids,  159 

mercury,  162 
Pyrometry,  optical,  276 

radiation,  276 

Quadrant  electrometer,  354 
Quality  of  sounds,  526,  532 
Quantity  of  heat,  226 

RaHiant  energy,  emission  of,  272 


Radiant  energy,  absorption  of,  272,  640 
Radiation,  269,  269-277,  553 

law  of,  635 

pyrometry,  276 
Radio-active  substances,  515 
Radiomicrometer,  423 
Radium,  504 
Radius  of  gyration,  64 
Rainbow,  602 

Ratio  of  Specific  Heats,  237,  284,  520 
Rays,  a,  0,  and  7,  504 

cathode,  490 

Rdntgen,  493 

Rectilinear  propagation  of  light,  554 
Reeds,  543 
Reflection  at  a  plane  surface,  572 

laws  of,  557,  573 

of  electromagnetic  waves,  474 
Reflection  of  light  waves,  572 

radiant  heat,  269 

of  sound  waves,  523 

of  waves,  188 

phase  change  in,  189,  190,  607 

selective,  668 

total,  599 
Refraction  of  light,  559,  582 

waves,  193 
Refractive  index,  582,  588 

measurements  of,  585,  586 
Regenerative  methods  of  liquefaction,  260 
Residual  charge,  362 
Resistance,  electrical,  391 

specific,  393 

standards  of,  395 

temperature  coefficient  of,  395 

thermometers,  395 

units  of,  392 

Resistances  in  series  and  parallel,  397 
Resistivity,  393 
Resolving  power,  614,  617 
Resonance,  540,  544 

electrical,  473 
Resonator,  Hertz,  473 
Resonators,  540 
Restitution,  coefficient  of,  116 
Resultant  of  forces  on  a  body,  68-  73 

on  a  particle,  35-39 
Reverberation,  547 
Reversal  of  spectral  lines,  641 
Reversible  processes,  285,  293 

efficiency  of,  289 
Rigidity,  simple.  113 
Ripples,  193-196 
Rods,  vibration  of,  543 
Roget's  spiral,  467 

Rdmer,    determination    of   the    velocity    of 
light,  562 


INDEX  TO  SUBJECTS 


691 


RSntgen  raya,  493 
Rotation,  6,  51-56,  61,  66 

of  plane  of  polarization,  663-666 
Rowland's  experiment  on  moving  charge, 

383 

measurement  of  the  mechanical  equiva- 
lent of  heat,  279 

Rumford's    experiment    on    the    nature    of 
heat,  200 

Saccharimetry,  665 

Sail-boat,  37 

Saturation  current  in  gases,  497 

curve,  245 
Scalar,  11 

Scales,  muscial,  531 
Screw,  98 
Second  of  time,  13 
Secondary  cells,  415 

current,  439 
Self-induction,  445 
Shadows,  554 
Shear,  110 

modulus,  114 
Shunts,  398 
Siemens  armature,  457 
Simple  harmonic  motion,  78-86 

composition  of,  178-181 

pendulum,  82 
Sines,  curve  of,  182 
Siphon,  160 

barometer,  150 
Siren,  529 
Sky  color,  645 
Solar  spectrum,  642 
Solids,  106,  109-120 

thermal  conductivity  of,  268 
Solenoid,  376,  379 
Solutions,  freezing  point  of,  244 
Sound,  diffraction  of,  525 

direction,  526 

interference  of,  535 

reflection  of,  523 

refraction  of,  524 

velocity  of,  519-521 
Spar,  Iceland,  649 
Spark  discharge,  489 
Specific  gravity,  108 

heat,  227 

measurement  of,  230 
water,  228 

heats,  difference  of,  of  gases,  236 
ratio  of,  of  gaaes,  237 

inductive  capacity,  364 

resistance,  393 

volume,  221,  250 
Spectr«,!270,  275,  586,  631-644 


Spectroscope,  627 
Spectrum,  586 

distribution  of  energy  in,  275,  635 
Speed,  11,  14 

Spherical  aberration,  580,  595-598 
Spring,  43,  79 
Stability  of  flotation,  134 
Standard  temperatures,  212 
Standards  of  luminosity,  631 
Statics,  6 

Stationary  waves,  190,  609 
Steam  engine,  295-300 
Stefan's  law,  274 
Storage  cell,  415 
Strain,  110 
Stress,  111 

Strings,  vibration  of,  536 
Sublimation,  252 
Superheating,  298 
Surface  tension,  139-  145 
Susceptibility,  426 

Tangent  galvanometer,  384 

law,  326 

Telegraphy,  wireless,  477 
Telephone,  462 
Telescope,  astronomical.  624 

Galilean,  624 

reflecting,  625 
Temperature,  202,  20* 

absolute  scale,  of,  290 

critical,  254 

gradient,  264 

standard,  202 

zero  of,  206 
Tension  of  vapor,  246 
Terrestrial  magnetism,  329-334 
Tesla  induction  coil,  450 
Theory,  kinetic,  108,  155,  201,  405 
Thermal  unit,  226 

conductivity,  264 
Therm o-dyuamic  surface,  259 
temperature,  290 

dynamics,  first  law  of,  281 
second  law  of,  288 

electric  power,  421 

electricity,  419-424 
Thermometer,  gas,  203-207 

maximum  and  minimum,  209 

mercurial,  207 
errors  of,  208 

platinum,  210 

thermoelectric,  211 
Thermometrio  scales,  202 
Thermometry,  212 
Thermopile,  271,  422,  633 
Thick  plates,  608 


692 


INDEX  TO  SUBJECTS 


Thin  plates,  605 
Thompson  effect,  422 

galvanometer,  386 
Tides,  103 
Time,  units  of,  13 
Tones,  quality,  526,  532 
Toepler's  electrostatic  machine,  357 
Torricelli's  experiment,  149 

theorem,  136 
Torsion,  114 

balance,  319,  348 

constant  of,  84 

pendulum,  84 
Total  reflection,  599 
Tourmaline,  652 
Transformers,  457 
Transition  layer,  601 
Translation,  6,  61,  66 
Translucency,  558 
Transmitter,  telephone,  462 
Transparency,  558 
Triangle  method,  9 
Triple  point,  253 
Tuning-fork,  544 
Turbines,  299 


Umbra,  555 
Uniaxial  crystals,  652 
Unit  of  capacity,  359,  365 

current,  378 

electromotive  force,  390 

force,  30 

heat,  226 

inductance,  447 

length,  8 

mass,  28 

pole  strength,  319 

potential,  352,  390 

power,  44 

quantity,  352,  378 

resistance,  392 

time,  13 

work,  42 
Units,  absolute  system  of,  104-105 

derived,  104 

electrical  and  magnetic,  480-482 

fundamental,  104 
Van  der  Waal's  law,153,  257 
Vaporization,  245 
Vaporization,  heat  of,  250 
Vapor  pressure  or  tension,  246 
Vector.  10 
Velocity,  angular,  51 

of  light,  562-566 

linear,  11-18,  59,  80 

outflow  of  liquid,  136 

•ound  in  air,  519 

in   gases,    effect    of   tempera- 
ture on,  521 


Velocity,  sound  in  liquids.  545 
Vibrations  of  bells,  544 

columns  of  gas,  539 

rods,  543 

strings,  536 
Viscosity,  130,  154 
Visibility,  557 
Vision,  619 
Voice,  535 
Volt,  391,  480 

Voltaic  cell,  371,  373,  411-419 
Voltameter,  411 
Voltmeter,  399 

Volume,  change  of,  during  fusion,  243 
Vowel  characteristics,  535 

Water,  expansion  of,  220 

specific  heat  of,  228 
Watt's  governor,  34 
Watt,  the,  44 
Wave,  electric,  470-480 

heat,  176 
Wave  amplitude,  182 

damped,  197,  478 

energy  and  intensity,  197 

frequency,  182 

front,  656 

length,  182,  640,  642 

motion,  173 

surface,  in  uniaxial  crystals,  651 
Waves,  interference  of,  184,  196 

longitudinal,  175,  188 

on  cords,  174,  186 

polarized,  183,  186 

reflection  of,  188,  193 

refraction  of,  193,  195 

stationary,  190 

transverse,  175 

on  surface  of  liquids,  191 
Wehnelt  interrupter,  449 
Weight,  31 
Weston  ammeter,  390 

cell,  415 

Wheatstone's  bridge,  210,  272,  400 
Wheel  and  axle,  95 
Wimshurst's  machine,  357 
Wireless  telegraphy,  477 
Work,  40-51,  63,  238,  238,  277 

by  a  piston,  129 

diagram  of,  42 

unit  of,  42 

Yard,  8 

Young-  Helmholtz  theory  of  color,  647 
Young's  experiment  on  light,  569 
modulus,  114 

Zero,  absolute,  290 

gas  thermometer,  204 
Zeeman  effect,  666 


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